数理解析研究所講究録 1341 巻 2003 年 12-30

12-Function Generalized Fractional Integration
Operators in Subclasses of Univalent Functions:
Some Distortion and Characterization Theorems$H$

Virginia S. Kiryakova 1
Megumi Saigo 2
Shigeyoshi Owa3

1. Introduction
Let $A(n)$ denote the class of functions of the form$f(z)=z+ \sum_{k=n+1}^{\infty}a_{k}z^{k}$(n $\in N=$ {1,2,3,$\ldots\})$,(1)which axe analytic in the unit disk $U=\{z : |z|<1\}$ , and let $S(n)$ denote the subclassof $A(n)$ of univalent functions in . The s0-called subclass of functions with negative
coefficients is also often considered, denoted by $T(n)\subset S(n)$ , of the functions of the form$U$$f(z)=z$$- \sum_{k=n+1}^{\infty}a_{k}z^{k}$with$a_{k}\geq 0$$(\ \ovalbox{\tt\small REJECT} n+1,n+2, \ldots)$.

(2)

We consider some mapping, distortion and other characterization properties of theoperators of the generalized fractional calculus involving Fox’s $H$-functions(Kiryakova[7] , in the classes $A(n),S(n),T(n)$ and their subclasses of the s0-called starlike and convex
functions of order , $0\leq\alpha<1$ .
$)$$\alpha$In this way we extend our previous results (see Kiryakova, Saigo and Owa [9]) related
to the operators of generalized fractional calculus involving Meijer’s Functions: andincluding the hypergeometric fractional integration operators by Saigo ([21]-[23], [31])and Hohlov $([3],[4])$ , the Appell’s -function operators by Saigo ([24],[25]) and mostof the classical integral operators considered in classes of univalent functions by various
authors.
$G$$F_{3}$

1 Istitute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia 1090, Bulgaria
2 Depart ment of Applied Mathematics, Fukuoka University, Fukuoka 814-0180Japan
3 Department of Mathematics, Kinki University, Higashi-Osaka Osaka 577 Japan13
2. Generalized Fractional Calculus Operators
We remind first the definitions of some special functions referred to in our paper.
By a Fox’s $H$ -function we mean ageneralized hypergeometric function defined bymeans of the Mellin-Barnes type contour integral$H_{p,q}^{m,n}[ \sigma|(a_{k},A_{k})_{1}^{p}(b_{k},B_{k})_{1}^{q}]=\frac{1}{2\pi i}\int_{\mathcal{L}}$$\frac{\prod_{k=1}^{m}\Gamma(b_{k}-B_{k}s)\prod_{j=1}^{n}\Gamma(1-a_{j}+sA_{\mathrm{j}})}{\prod_{k=m+1}^{q}\Gamma(1-b_{k}+sB_{k})\prod_{j=n+1}^{p}\Gamma(a_{j}-sA_{j})}\sigma^{s}ds$

,

,

(3)

where is asuitable contour in , the orders $(m,n,p, q)$ are integers $0\leq m\leq q,0\leq n\leq$$R,B_{k}$0,1, , ,
0,and the parameters$=1$ ,, are such that $A_{j}(b_{k}+l)\neq B_{k}(a_{\mathrm{j}}-l^{J}-1)$ , , $l’=0,1)$ , . For various type of contours and conditions for existence and analyticity of function (3) in disks, one can see [14],[28],[7, App.], etc.
whose radii are
When $A_{1}=\ldots=A_{p}=1$ , $B_{1}=\ldots=B_{q}=1$ , (3) turns into the more popular, Ch.5],[14],[7]). The G- and $H$-functions encompass
Meijer’s -fimction (see [2,almost all the elementary and special functions and this makes the knowledge on themvery useful. Observe that the generalized hypergeometric functions $pqF$ are special cases$C$$\mathcal{L}’$$p$$a_{j}$$k$$R,A_{j}$

$\in$

$>$

$j$

$=$

$\ldots$$p$$b_{k}$

$>$

$\in$

$l$

$\ldots$$q$$\ldots$$\rho=\Pi_{j=1}^{p}A_{j}^{-A_{j}}\Pi_{k=1}^{q}B_{k}^{B_{\mathrm{k}}}>0$$\subset C$$G$$\mathrm{V}\mathrm{o}\mathrm{I}.\mathrm{I}$of the G-function:,$1-a_{p}]$$\ldots,1-b_{q}1-a_{1},\ldots,$$pF_{q}(a_{1}, \ldots, a_{\mathrm{P}};b_{1}, \ldots, b_{q};\sigma)=\frac{\prod_{j_{-}^{-}1}^{q}\Gamma(b_{j})}{\prod_{j=1}^{p}\Gamma(a_{j})}G_{p,q+1}^{1,p}[-\sigma|01-b_{1},$(appearing as solutions of fractional order differwhile the Mittag-Lefller functions
ential and integral equations) and the Wright’s generalized hypergeometric functions
with irrational $A_{j},B_{k}>0$ , give examples of $H$ -functions, not reducible to G-functions:$E_{\rho,\mu}$$\mathrm{p}q\Psi$$pq\Psi$($(a_{1},A_{1}),\ldots,(a_{p},A_{p})(b_{1},B_{1}),\ldots,(b_{q},B_{q})$;$\sigma)=\sum_{k=0}^{\infty}\frac{\Gamma(a_{1}+kA_{1})\ldots\Gamma(a_{p}+kA_{p})}{\Gamma(b_{1}+kB_{1})\ldots\Gamma(b_{q}+kB_{q})}\frac{\sigma^{k}}{k!}$(4)$=H_{p,q+1}^{1,p}[-\sigma|(0,1),(1-b_{1},B_{1}),\ldots,(1-b_{q},B_{q})(1-a_{1}, A_{1}), \ldots,(1-a_{p},A_{p})]$.

However, for $A_{1}=\ldots=A_{p}=B_{1}=\ldots=B_{q}=1$ ,$\mathrm{r}^{\Psi_{q}(}(a_{1},1),\ldots,(a_{p},1)(b_{1},1),$$\ldots,(b_{q},1)$;.

$\sigma)=[\frac{\prod_{j_{-}^{-}1}^{p}\Gamma(a_{j})}{\prod_{\dot{*}=1}^{q}\Gamma(b_{\dot{t}})}]\mathrm{P}F_{q}(a_{1}, \ldots, a_{p};b_{1}, \ldots,b_{q};\sigma)$(5)

In this scheme of $H$-functions we have recently included and studied also multi-index, called multiindex Mittag-Leffler functions (see Kiryakova [8]):analogues of$E_{\rho,\mu}$$E(\begin{array}{l}\underline{1}\rho i\end{array})$,$( \mu.\cdot)(z)=\sum_{k=0}^{\infty}\varphi_{k}z^{k}=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(\mu_{1}+k/\rho_{1})\ldots\Gamma(\mu_{m}+k/\rho_{m})}$

.

(6)

14
are arbitrary real numbers,
Here $m>1$ is an integer, , , $\rho_{m}>0$ and , ,and for $m=1$ one gets the classical Mittag-Leffler function. In terms of the H- andfunctions,$\rho_{1}$$\ldots$$\mu_{1}$$\mu_{m}$$\ldots$$pq\Psi$$E(\begin{array}{l}\underline{1}\rho_{|}\end{array})$,$(\mu_{*}.)(z)=1\Psi m((\mu_{i_{i}^{\frac{1}{\rho}}},)_{1}^{m}(1,1)$; $z)=H_{1,m+1}^{1,1}[-z|(0,1)$ ,$(1- \mu_{i}, \frac{1}{\rho})_{1}^{m}(0, 1)i]$.

(7)

Using as kernel-function aMeijer’s -function, and more generally-aFox’s H-functionof peculiar order $(m, 0,m,m)$ , ageneralized ffactional calculus has been developed in
Kiryakova [7] that includes as special cases almost all the known operators of fractional
integration and differentiation studied by many authors. Especially, even the particularcase with a -function kernel, has been shown (Kiryakova [7, Ch.5], Kiryakova, Saigo
and Owa [9], Kiryakova, Saigo and Srivastava [10] to encompass most of the integr0differential operators already popular in univalent functions theory.
$G$

$G$

$)$

$>0$ , $i=1$ ,, $m$ . We consider
Let $m\geq 1$ be an integer;
as amultiorder of fractional integration, resp.,
as multiweight,
as additional parameter. The integral operators defined asfollows:$\delta_{i}\geq 0.\gamma_{i}\in R,\beta\dot{.}$$\delta=$$\ldots$$\gamma=(\gamma_{1}, \ldots, \gamma_{m})$$(\delta_{1}, \ldots,\delta_{m})$$\beta=(\beta_{1}, \ldots,\beta_{m})$,$\int_{0}^{1}H_{m,m}^{m,0}[\sigma|(\gamma.\cdot+.\delta.\cdot+1\frac{1}{\beta}.\cdot’\frac{1}{\beta_{i}},)_{1}^{m}(\gamma_{*}+1-\frac{-1}{\beta_{i}},\frac{1}{\beta})_{1}^{m}]f(z\sigma)d\sigma$$\mathrm{i}\mathrm{f}.\cdot\sum_{=1}^{m}\delta_{\dot{*}}$$I_{(\beta_{i}),m}^{(\gamma_{\dot{1}}),(\delta.)}.f(z)=\{$$f(z)$,

if$\delta_{1}=\delta_{2}=\ldots=\delta_{m}=0$

$>0$

,

(8)

,

axe said to be multiple ( -tuple)Erd\’elyi-Kober fractional integration operators and more$m$generally, all the operators of the formIwithare called briefly generalized ( -tuple)ffactional integrals.
$f(z)=z^{\delta_{0}}I_{(\beta_{*}),m}^{(\gamma.),(\delta.)}.\cdot.f(z)$$\delta_{0}\geq 0$,$m$classical Riemann-Liouville derivative. For $m=1$ , operators (8) turn into the Erd\’elyi, widely used in the applied mathematical analysis (seeKober fractional integrals$I_{\beta}^{\gamma,\delta}$$[26],[\dot{‘}])$and to the $class:cal$ Riemann-Liouville fractional integrals$\Gamma_{\beta’}^{\delta}f(\approx)=\int_{0}^{1}\frac{(1-\sigma)^{\delta-1}}{\Gamma(\delta)}\sigma^{\gamma}f(z\sigma^{1/\beta})d\sigma$$R^{\delta}$:$(\delta>0,\gamma\in R, \beta>0)$$R^{\delta}f(z)=z^{\delta} \int_{0}^{1}\frac{(1-\sigma)^{\delta-1}}{\Gamma(\delta)}f(z\sigma)d\sigma=z^{\delta}F_{1’}^{\delta}f(z)$$(\delta>0)$

,

,
(9)

namely:$R^{\delta}f(z)=z^{\delta}I_{1,1}^{0,\delta}f(z)$,$I_{\beta}^{\gamma,\delta}f(z)=I_{1,1}^{\gamma,\delta}f(z)$,for $m=2$ -into the hypergeometric fractional integrals (Love, Saigo, Hohlov, etc.), andfor various other special choices of $m\geq 1$ and of parameters, to many other generalized
integration and differentiation operators, used in analysis, including in univalent functions
theory, integral transforms and special functions, differential and integral equations, etc.

15
The main feature of the generalized ( -tuple)fractional integrals is that single integrals(8) involving $H$ -functions(or -functions in the simpler case of equal $\beta_{i}=\beta>0$ , $i=$, , ) can be equivalently represented by means of commutative compositions finite
number (m) of Erd\’elyi-Kober integrals (9), namely: in the case considered here, for, $i=1$ , , ,
$m$

$G$

$1$

$\ldots$$m$$\gamma_{i}\geq-1,$ $\delta_{i}\geq 0,\beta_{i}>0$$m$$\ldots$$I_{(\beta_{i}),m}^{(\gamma.),(\delta.)}..f(z)=[ \prod_{i=1}^{m}I_{\beta}^{\gamma,\delta_{i}}]f(z)$

(10)

.

$= \int_{0}^{1}\cdots\int_{0}^{1}[\prod_{i=1}^{m}\frac{(1-\sigma_{i})^{\delta.-1}\sigma^{\gamma}}{\Gamma(\delta_{i})}.\dot{.}.]f(z\sigma^{\frac{1}{1\rho_{1}}}\ldots\sigma_{m}^{\mathrm{f}^{1}})\mathrm{f}1d\sigma_{1}\ldots d\sigma_{m}$If some of the are zeros: $\delta_{1}=\ldots=\delta_{\epsilon}=0,1\leq s\leq m$, the corresponding multipliers
and the multiplicity of (8),(10) reduces from to $m-s$
are identity operators(the same for the order of the kernel $H$-functions). Decomposition (10) is the key tonumereous applications of (8), arising from the simple but quite effective tools of the Gand H-functions.
Adetailed theory, called generalized fractional calculus and an analogue of the classicalfractional calculus and its different applications are proposed in [7], Here we consider
some mapping properties of operators (8) in classes of analytic functions in the unit disk$\delta_{i}$$m$$(I_{\beta_{\dot{1}}}^{\gamma j}’\delta.\cdot=I)$$U=\{z:|z|<1\}$ .

Using only the simple properties of Fox’s $H$ -function([14],[28],[7, App.]), one easily
obtains the following.
$(i=1, \ldots,m)$ , and each $p> \max_{i}[-\beta_{i}(\gamma_{i}+1)]$ ,Lemma 0. For$\delta_{;}$$\geq 0,\gamma_{i}\in R,\beta_{i}>0$$I_{(\beta_{}),m}^{(\gamma.),(\delta\dot{.})}.\{z^{p}\}=\lambda_{p}z^{p}$

Then the conditions$\delta.\cdot\geq 0$

,

with

$\lambda_{p}=\prod_{i=1}^{m}\frac{\Gamma(\gamma_{i}+1+p/\beta)}{\Gamma(\gamma_{}+\delta_{i}+1+p\beta_{i})}i>0$, $i=1$ ,$\gamma_{i}\geq-1,\beta_{i}>0$

ensure that (11) holds for each $p\geq 0$ , $i.e$ . in the class$\ldots$

$A$

,

$m$

.

,

(11)
(12)

and its subclasses.

-image of an arbitrary power function $f(z)=z^{p}$ , we$Proof$. . To evaluate theuse an extension of known integral formulas for the $H$-functions, namely formula (E.21),$I_{(\beta i),m}^{(),(\delta)}\gamma$[7, App.]:,$\int_{0}^{1}H_{m,m}^{m,0}[\sigma|(a_{j},C_{J})_{1}^{m}(b_{i},c_{i}^{i})_{1}^{m}]d\sigma=\prod_{i=1}^{m}\frac{\Gamma(b_{i}+C_{i})}{\Gamma(a_{i}+C_{\dot{1}})}$

for $a_{i}>b_{:}>-C_{i}(i=1, \ldots,m)$ .

Then, according to the well known $H$ -function’sproperty (see (E.9), [7, App.]), we obtai$I_{(\beta i_{),m}^{),(\delta_{})}\{z^{p}\}=\int_{0}^{1}H_{m,m}^{m,0}}^{\mathrm{t}\gamma}[ \sigma|(\gamma_{j}+\delta_{i}+1.\cdot\frac{1}{\beta}\dot{.},\frac{1}{\beta m’ 1}.\cdot,)_{1}^{m}(\gamma_{i}+1-\frac{-1}{\beta},\frac{1}{\beta})]z^{p}\sigma^{p}d\sigma$,$=z^{p} \int_{0}^{1}H_{m,m}^{m,0}[\sigma|(\gamma_{i}+\delta_{\dot{1}}+1+(p-1)/.\beta_{i})_{1}^{m}(\gamma_{\dot{1}}+1+(p-1)/\beta.)_{1}^{m}]d\sigma=z^{p}\prod_{=1}^{m}.\frac{\Gamma(\gamma_{i}+1+p/\beta)}{\Gamma(\gamma.+\delta_{\dot{1}}+1+p\beta_{})}i=\lambda_{\mathrm{p}}z^{p}$16
where the conditions
and $\gamma_{i}>-1-p/\beta_{i}(i=1,$\ldots ,m), i.e. p, it suffices to ask,p.

$\gamma_{\dot{\mathrm{f}}}+\delta_{i}+p/\beta_{\dot{v}}>\gamma_{i}+p/\beta>-1$$\delta_{i}\geq 0$$z^{p}$$\geq 0$(i $=1,$\ldots , m) are ensured by. To have (11) for all$> \max_{i}[-\beta_{i}(\gamma_{\dot{0}}+1)]$$\gamma_{i}\geq-1$$\blacksquare$

In view of formula (11), for considering functions in the classes $A(n)$ , $S(n)$ , $T(n)$ ,it is suitable to nomalize the operators (8) by the multiplier constant $[\lambda_{1}]^{-1}(p=1)$ .
Therefore, further we consider the generalized fractional integrals (using the same name forthe normalized
stressing this fact by an additional “tilde” in the denotation:,$\tilde{I_{(\beta_{i}),m}}^{(),(\delta_{})}\gamma i:=[\lambda_{1}]^{-1}I_{(\beta.),m}^{(\gamma_{\triangleleft}),(\delta\dot{.})}.\cdot)$$\tilde{I_{(\beta.),m}}^{(\gamma_{i}),(\delta_{i})}.f(z)$$:= \prod_{i=1}^{m}\frac{\Gamma(\gamma\dot{.}+\delta_{i}+1+1/\beta_{i})}{\Gamma(\gamma_{j}+1+1/\beta_{i})}I_{(\beta i),m}^{(),(\delta_{j})}\gamma jf(z)$

.

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Thus, from Lemma 0and the more general results in [7, Ch.5, \S 5.5], [11, Th.l], we caneasily obtain the following:

Theorem 1.

Under the$pa\mathit{7}umeters^{f}$$\delta_{i}\geq 0$,

conditions (12):$\gamma_{i}>-1$,

the generalized fractional integralof a power series (1) has the form$\tilde{I_{(\beta.),m}}^{(\gamma),(\delta.)}.\cdot$$(i=1, \ldots, m)$$\beta_{i}>0$

maps the class $A(n)$ into itself, and the image$\tilde{I}f(z)=I_{(\beta),m}\tau_{\mathrm{Y}i}),(\delta.\cdot)\{z+\sum_{k=n+1}^{\infty}a_{k}z^{k}\}=z+\sum_{k=n+1}^{\infty}\theta(k)a_{k}z^{k}\in A(n)$

,

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with multipliers’ sequence:$\theta(k)=\prod_{=1}^{m}\frac{\Gamma(\gamma_{i}+1+k/\beta_{})\Gamma(\gamma_{i}+\delta_{i}+1+1/\beta_{i})}{\Gamma(\gamma_{\dot{2}}+\delta_{i}+1+k/\beta_{i})\Gamma(\gamma_{i}+1+1/\beta_{i})}>0$

Pr oof.

(k$=n+1,n+2,$\ldots ).

(15). First we need to establish the fact that$\lim_{k\prec\infty}|\theta(k)|^{1/k}=1$

.

(16)

Denote, for brevity in the proofs of Th. 1and next Th. 2,$a_{i}=\gamma_{i}+\delta_{i}+1$,$b\dot{.}$$=\gamma:+1$,$k/\beta_{:}=\kappa_{i}$,and additionally, $c_{i}=a_{i}+(n+1)/\beta_{:},d_{i}=b_{j}+(n+1)/\beta_{\dot{1}}$ , i $=1$ , \ldots , m,from where and from (12) evidently,$a_{i}\geq b_{:}$, C:$\geq d_{i}$, i $=1$ ,\ldots ,m and$\kappa_{i}arrow\infty$

as k$arrow\infty$

The known asymptotics$\frac{\Gamma(b+\kappa)}{\Gamma(a+\kappa)}\sim k^{b-a}$as$\kappaarrow\infty$,

yields$[ \frac{\Gamma(b.+\kappa)}{\Gamma(a_{}+\kappa)}.]^{1/k}\sim(\kappa^{-\delta:})^{1/k}$$=(k^{1/k})^{-\delta}:\cdot(\beta_{}^{\delta_{i}})^{1/k}$

.

(17)

17
and the limit equalities
give:$\lim_{karrow\infty}k^{1/k}=1,\lim_{k\prec\infty}(k^{1/k})^{p}=1,\lim_{karrow\infty}q^{1/k}=1$$n. arrow\infty!\mathrm{i}\mathrm{m}[\frac{\Gamma(b.+\kappa_{\dot{1}})}{\Gamma(a_{i}+\kappa\dot{.})}]^{1/k}=1$

and

for ,
$p$

$\lim_{karrow\infty}[\frac{\Gamma(a_{i}+1/\beta_{i})}{\Gamma(b_{i}+1/\beta_{i})}]^{1/k}=\lim_{karrow\infty}q^{1/k}=1$$q=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$,.

We have then$\lim_{k\prec\infty}|\theta(k)|^{1/k}=\lim_{karrow\infty}\prod_{i=1}^{m}[\frac{\Gamma(b_{*}+\kappa_{*})}{\Gamma(a_{*}+\kappa_{i})}..\cdot]^{1/k}[\frac{\Gamma(a_{}+1/\beta_{i})}{\Gamma(b_{\dot{1}}+1/\beta_{})}]^{1/k}=1$, i.e. (16).

In order to have (11) valid with $p\geq 0$ , we require conditions (12). Then,$\tilde{I_{(\beta_{i}),m}}^{(\gamma_{i}),(\delta_{i})}\{z\}=z$and$\tilde{I_{(\beta),m}}^{(\gamma_{i}),(\delta_{})}\{z^{k}\}=\frac{\lambda_{k}}{\lambda_{1}}z^{k}=\theta(k)z^{k}$and term-by-term integration of power series (1) gives series (14). By virtue of the CauchyHadamard formula, the radius of convergence of the first series, as analytic function in, and that of the latter series is calculated by
the unit disk, is$R=\{\varlimsup_{karrow\infty}|a_{k}|^{1/k}\}^{-1}\geq 1$$\tilde{R}=\{\varlimsup_{karrow\infty}|a_{k}|^{1/k}\cdot|\theta(k)|^{1/k}\}^{-1}$,given by series (14) is analytic function in theand the image
therefore, series with positiveof the multipliers
positivenessunit disc, too. Note that due to(like in $A(n)$ ) and negative (like in $T(n)$ ) coefficients map into series of same kind.
$\tilde{R}\geq 1$$\tilde{I}_{(\beta_{}),m}^{(\gamma)_{1}(\delta_{j})}f(z)$$\theta(k)$$\blacksquare$

The Hadamard product (convolution) of two analytic functions in$f(z)= \sum_{k=0}^{\infty}a_{k}z^{k}$,$U$$g(z)= \sum_{k=0}^{\infty}b_{k}z^{k}$is defined by$f*g(z):= \sum_{k=0}^{\infty}a_{k}b_{k}z^{k}$.

Theorem 2. In the class $A(n)$ the generalizedfractional integral (13) can be represented
by the Hadamard product(18)$=h(z)*f(z)$ ,$\tilde{I_{(\beta),m}}^{(\gamma.),(\delta_{i})}.f(z)$where the function $h(z)\in A(n)$ is expressed by the Wright generalized hypergeometricfunction (4):$h(z)=zf \sum_{k=n+1}^{\infty}\theta(k)z^{k}$$=z+z^{n+1}.

\prod_{*=1}^{m}\Gamma(\gamma_{\dot{1}}+\cdot 1\mathrm{i}\Gamma(\gamma_{i}+\delta.+1+1/\beta)+1/\beta_{\dot{1}})m+1\Psi m((1,1), (\gamma.\cdot+1+(n+1)/\beta_{}.’

1/\beta_{i})_{1}^{m}(\gamma_{\dot{1}}+\delta_{j}+1+(n+1)/\beta.,1/\beta_{i})_{1}^{m}$;$z)_{1\cap\backslash },\cdot$

18
Proof.
,In the expression for $h(z)$ we change the index of summation k, via k $=j+(n+1)$ , and using the short denotations
to j $=0,$ 1,$=$$n+1$ $n+2$,$\infty$\ldots ,\ldots$\infty$in (17), we get$h(z)=z+ \sum_{k=n+1}^{\infty}\theta(k)z^{k}=z$$+z^{n+1}[ \lambda_{1}]^{-1}\sum_{j=0}^{\infty}\lambda_{j+(n+1)}z^{j}$$=z+z^{n+1}[ \lambda_{1}]^{-1}\sum_{j=0}^{\infty}\Gamma(1+j)\prod_{\dot{*}=1}^{m}\frac{\Gamma(d_{i}+j/\beta_{i})}{\Gamma(c_{i}+j/\beta_{i})}\cdot\frac{z^{j}}{j!}$$=z+z^{n+1}[\lambda_{1}]_{m+1}^{-1}\Psi_{m}$(;$(1, 1),(d_{1},1/\beta_{1}),$$\ldots, (d_{m}, 1/\beta_{m})(c_{1},1/\beta_{1}), \ldots,$$(c_{m},1/\beta_{m})$

$z$

),

which gives (19).

$\blacksquare$

Corollary 1.
In the case $n=1_{f}$ in the classes , $S,T$ the representation of the“convolution function” $h(z)$ in (18) simplifies as follows:$A$$h(z)=z[\lambda_{1}]_{m+1}^{-1}\Psi_{m}((1,1), (\gamma_{i}+1+1/\beta_{i},1/\beta_{*}.)_{1}^{m}(\gamma_{i}+\delta_{i}+1+1/\beta_{i},1/\beta\dot{.})_{1}^{m}$; $z)$ .

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Corollary 2.
When all $\beta_{i}=\beta>0$ , $i=1$ , , , and especially for shortness ofdenotations it is taken $\beta=1$ , for the generalized fractional integrals with Meijer’s Gfunction in the kernel,$\ldots$$m$$\tilde{I}_{1,m}^{(\gamma.),(\delta_{i})}.f(z)=\tilde{I}_{(1,1,\ldots,1),m}^{(\gamma),(\delta_{})}f(z)=[\lambda_{1}]^{-1}\int_{0}^{1}G_{m,m}^{m,0}[\sigma|(\gamma_{i}+\delta_{\dot{1}})_{1}^{m}(\gamma_{i})_{1}^{m}]f(z\sigma)d\sigma$

we get respectively the simpler representations of multipliers’ sequencetion function $h(z)$ as follows:$\theta(k)=\prod_{i=1}^{m}\frac{(\gamma_{i}+2)_{k-1}}{(\gamma_{j}+\delta_{*}+2)_{k-1}}.>0$$\theta(k)$

,

(21)

and convolu-(22)$(k =n+1,n+2, \ldots)$with $(a)_{k}=\Gamma(a+k)/\Gamma(a)$ denoting the known Pochhammer symbol, and$h(z)=z+. \prod_{*=1}^{m}\frac{(\gamma_{i}+2)_{n}}{(\gamma_{}+\delta_{i}+2)_{n}}z^{n+1}m+1F_{m}((\gamma_{i}+\delta_{i}+2+n)_{1}^{m}1,(\gamma_{}+2+n)_{1}^{m}$

For $n=1$ ($i.e$ . in the classes$A$; $z)$ .

(23), $S,T$), $h(z)$ simplifies to $a_{m+1}F_{m}$ generalized hypergeO-

metric function:$h(z)=z_{m+1}F_{m}((\gamma_{i}+\delta_{}+2)_{1}^{m}1,(\gamma_{}+2)_{1}^{m}$; $z)$ .

(24)

Many special cases of operators (13), or of their modified formwith, especially in the case with kernel-function reducing to Meijer’scand$cz^{\delta_{0}}\tilde{I}_{(\beta i_{),m}^{),(\delta_{j})}f(\chi)}^{(\gamma}$$=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$$\delta_{0}\geq 0$

19-function, have been used very often in the univalent function theory, like the known
operators of: Biernacki, Komatu, Libera, Rusheweyh, Owa and Srivastava, Carlson andShaffer, Saigo, Hohlov, etc. (see the examples in [7, Ch.5], and details in Kiryakova, Saigo,
Owa [9], Kiryakova, Saigo, Srivastava [10] . Thus, the results below give as corollaries
$G$

$)$

corresponding properties of all these operators.

3. Distortion Inequalities in the Classes$S_{\alpha}(n)$and$L_{\alpha}(n)$Afunction $f(z)$ belonging to $S(n)$ is said to be starlike of order $\alpha(0\leq\alpha<1)$ if andonly if it satisfies the inequality$\Re\{\frac{zf’(z)}{f(z)}\}>\alpha$and this subclass is denoted by$\alpha(0\leq\alpha<1)$ if and only if$S_{\alpha}(n)$. Further, $f(z)\in S(n)$ is said to be convex of 0 rder

$K_{\alpha}(n)$(26)(z $\in U)$$\Re\{(1+\frac{zf’(z)}{f’(z)}\}>\alpha$and the subclass is denoted by, and also for any $0\leq\alpha<1$ ,(25)(z $\in U)$. We note that $f(z)\in K_{\alpha}(n)$ if and only if $zf’(z)\in$$S_{\alpha}(n)$$S_{\alpha}(n)\subseteq S_{0}(n)$,$K_{\alpha}(n)\subseteq K_{0}(n)$and$K_{\alpha}(n)\subset S_{\alpha}(n)$.

have been recently studied by Srivastava, Owa and Chatand
The classes,
terjea [30]. For $n=1$ , these denotations are usually used as, which are introduced earlier by Robertson [19]. Especially, taking $\alpha=0$ , we obtain
and $K$ of starlike and convex functions in , respectively.
the well-known classes
In the class $T(n)$ of functions (2) with negative coefficients, we take now the respective
intersections for $0\leq 0t$ $<1$ , $n\in N$ :$S_{\alpha}(n)$$K_{\alpha}(n)$$S_{\alpha}(1)=S^{*}(\alpha)$$K_{\alpha}(1)=$$K(\alpha)$$U$$S^{*}$$T_{\alpha}(n)=S_{\alpha}(n)\cap T(n)$,$L_{\alpha}(n)=K_{\alpha}(n)\cap T(n)$

.

(27)

The latter classes were considered by Chatterjea [1] and in particular, case $n=1$ gives,, [27].
the Silverman’s classes
For functions of these classes we propose here some distortion inequalities in terms ofthe generalized fractional calculus operators (13).
We need first the following lemmas given by Chatterjea [1].
ifLemma 1. Let the function $f(z)$ be defined by (1). Then $f(z)$ is in the classand only if$T^{*}(\alpha)$$L(\alpha)$$T_{\alpha}(n)$$\sum_{k=n+1}^{\infty}\frac{k-\alpha}{1-\alpha}a_{k}\leq 1$

.

(28)

20
Let the function $f(z)$ be defined by (1). Then $f(z)$ is in the class

Lemma 2.
and only if$\sum_{k=n+1}^{\infty}\frac{k(k-\alpha)}{1-\alpha}a_{k}\leq 1$.

$L_{\alpha}(n)$

if

(29)

Applying Lemma 1and Theorem 1, we obtain Theorem 3.

satisfied and the function $f(z)$ defined by (1) belong to the classand $z\in U$ :inequalities hold for each

Let conditions (12) be$T_{\alpha}(n)$. Then the following$n\geq 1$$|\tilde{I_{(\beta_{\dot{*}}),m}}^{(),(\delta.)}\gamma j\cdot f(z)|\geq|z|$an$- \frac{1-\alpha}{n+1-\alpha}\theta(n+1)|z|^{n+1}$

(30)

,

(31)

$d$

$| \tilde{I_{(\beta_{i}),m}}^{(),(\delta.)}\gamma i.f(z)|\leq|z|+\frac{1-\alpha}{n+1-\alpha}\theta(n+1)|z|^{n+1}$

where the multiplier $\theta(n+1)$ is defined as in (15) namely:$\theta(n+1)=\prod_{i=1}^{m}.\frac{\Gamma(\gamma_{i}+1+(n+1)/\beta_{i})\Gamma(\gamma_{i}+\delta.+1+1/\beta_{i})}{\Gamma(\gamma.+\delta_{i}+1+(n+1)/\beta_{i})\Gamma(\gamma_{*}+1+1/\beta\dot{.})}.\cdot>0$

.

(32)

Equalities in (30) and (31) are attained by the function$f(z)=z- \frac{1-\alpha}{n+1-\alpha}z^{n+1}$

.

(33)

Theorem 4. Let conditions (12) be satisfied and the function $f(z)$ definied by (1) belong. Then the following inequalities hold for each $n\geq 1$ and $z\in U$ :to the class$L_{\alpha}(n)$$| \tilde{I}_{(\beta_{}),m}^{(),(\delta)}\gamma jf(z)|\geq|z|-\frac{1-\alpha}{n+1-\alpha}\frac{\theta(n+1)}{n+1}|z|^{n+1}$

(34)

,

(35)

and
$| \tilde{I_{(\beta_{}),m}}^{(\gamma),(\delta_{})}f(z)|\leq|z|+\frac{1-\alpha}{n+1-\alpha}\frac{\theta(n+1)}{n+1}|z|^{n+1}$

where the multiplier $\theta(n+1)$ is defined as in (32). Equalities in (34) and (35) are attainedby the function$f(z)=z- \frac{1-\alpha}{(n+1)(n+1-\alpha)}z^{n+1}$ .
(36)$ofTheoremsS$, . The main point in this proof is that the multiplier.

$Pr$ $oof$$l$is nonicreasing for $k\geq n+1$ . To verify this, let us start from the known
function
digamma-function$\theta(k)$$\Psi(x)=\Gamma’(x)/\Gamma(x)$,increasing for all x $>0$ .

21(Ii $(x)>0$ for all $x\neq-n$ , follows for example, from the representation of
p.723,(4) .) Then,$\Psi^{(n)}(x)$, [13,$x>0,\epsilon$

$>0$

$]$

$\Psi(x+\epsilon)=\frac{\Gamma^{l}(x+\epsilon)}{\Gamma(x+\epsilon)}>\frac{\Gamma’(x)}{\Gamma(x)}=\Psi(x)$, for$\epsilon$$>0$ ,

or, for the auxiliary function$\tilde{\Gamma}(x):=\Gamma(x+\epsilon)/\Gamma(x)$Then,$\tilde{\Gamma}(x)$$\Rightarrow$$\tilde{\Gamma}^{J}(x)=\frac{\Gamma’(x+\epsilon)\Gamma(x)-\Gamma(x+\epsilon)\Gamma’(x)}{\Gamma^{2}(x)}>0$

whenever$x\geq y>0$.

, $y|arrow b:+k/\beta_{j}$ (according to the notations in (17) and, $m$$>0)$ , gives for each $i=1$ ,This, for$a_{\dot{1}}$.

is also an increasing function, and so,$\frac{\Gamma(’x+\epsilon)}{\Gamma(x)}\geq\frac{\Gamma(y+\epsilon)}{\Gamma(y)}$$\geq b_{\dot{1}}$

,

$\Xi$

$\vdash\nu 1/\beta_{i},x\vdash+a_{i}+k/\beta_{\dot{1}}$$\ldots$$. \frac{\Gamma(a.+(k+1)/\beta_{i})}{\Gamma(a_{i}+k/\beta_{i})}\geq\frac{\Gamma(b_{\tilde{1}}+(k+1)/\beta_{i})}{\Gamma(b_{}+k/\beta_{\dot{1}})}$

therefore the required nonicreasing property for,follows:$\theta(k)$

,

(37)

$\frac{\theta(k)}{\theta(k+1)}=\prod_{=1}^{m}\frac{\Gamma(b_{i}+k/\beta_{i})}{\Gamma(b_{i}+(k+1)/\beta_{i})}\cdot\frac{\Gamma(a_{i}+(k+1)/\beta_{j})}{\Gamma(a_{}+k/\beta_{j})}.\geq 1$

Hence,$0<\theta(k)\leq \mathrm{E}\mathrm{t}(n+1)$

for each $k\geq n+1$ ,$(37’)$

and for $f(z)$ of form (2),$| \tilde{I}_{(\beta.),m}^{(\gamma),(l.)}.f(z)|\geq|z|-|\sum_{k=n+1}^{\infty}\theta(k)a_{k}z^{k}|$$\geq|z|-\theta(n+1)|z|^{n+1}\sum_{k=n+1}^{\infty}a_{k}\geq|z|-\theta(n+1)|z|^{n+1}\frac{1-\alpha}{n+1-\alpha}$,since in view of Lemma 1(see (28)), we have also$\sum_{k=n+1}^{\infty}a_{k}\leq\frac{1-\alpha}{n+1-\alpha}$.

Thus, inequality (30) is obtained. Next inequality (31) can be proved similarly andTheorem 4follows in analogous way by application of Lemma 2.

$\blacksquare$

Corollary 3.

If we set n $=1$ andconvex functions in U, respectively$\alpha=0$, we obtain for the subclasses of starlike and$f\in S^{*}\cap T(1)\Rightarrow|\tilde{I}f(z)|\geq|z|$ $- \frac{\theta(2)}{2}|z|^{2}$,$| \tilde{I}f(z)|\leq|z|+\frac{\theta(2)}{2}|z|^{2}$$f \in K\cap T(1)\Rightarrow|\tilde{I}f(z)|\geq|z|-\frac{\theta(2)}{4}|z|^{2}$,$| \tilde{I}f(z)|\leq|z|+\frac{\theta(2)}{4}|z|^{2}$

22
with multiplier.

$\theta(2)=\prod_{i=1}^{m}\frac{\Gamma(\gamma_{i}+1+2/\beta_{i})\Gamma(\gamma_{i}+\delta_{i}+1+1/\beta_{i})}{\Gamma(\gamma_{i}+\delta_{i}+1+2/\beta_{i})\Gamma(\gamma_{i}+1+1/\beta_{i})}$in (32) and subindices in
The case $m=1$ (simply omitting the sign
parameters) gives respective estimates for the classical Erd\’elyi-Kober operators (9).
Corollary 4.

$\prod m$

$i$

$i=1$

As applications of the above general results, we can derive the same kind ones for theoperators by Saigo ([21]-[23],[31]), and by Hohlov $([3],[4])$ as well as for the ffactional-function, recently studied by Saigo etintegrals and derivatives involving the AppelP
al. [24],[25]. All these cases fall in the scheme of the -function generalized ffactional
calculus operators (21) and the details are given in Kiryakova, Saigo, Owa [9].
$F_{3}$$\mathrm{s}$$G$

4. Characterization Theorems in the Classes

and $K(n)$ Now we consider$S^{*}(n)$some sufficient conditions for the operators of generalized fractional calculus to produce
starlike and convex functions. Namely, we denote by
the subclass of $A(n)$ of func, i.e. $S^{*}(n):=S_{0}(n)$ . Analogously, $K(n):=K_{0}(n)$ is thetions satisfying (25) with$f(z)$$A(n)$
satisfying (26) with
subclass ofof functions.
fitvm Silverman’s results [27], one can formulate the following auxiliary lemmas.
$S^{*}(n)$$\alpha=0$$\alpha=0$

If the function $f(z)$ defined by (1) satisfies the condition.
,

Lemma 3.

(38)$|a_{k}|\leq 1$$\sum_{k=n+1}^{\infty}h$then $f(z)\in S^{*}(n)$ . The equality in (38) is attained by the function$g_{1}(z)$ $=z+ \epsilon(n+1)\sum_{k=n+1}^{\infty}\frac{z^{k}}{k^{2}(k+1)}$,$\epsilon$$=comt$, $|\epsilon|=1$ ,.

$z\in U$(39)

If the function $f(z)$ defined by (1) satisfies the condition

Lemma 4.

$\sum_{k=n+1}^{\infty}k^{2}|a_{k}|\leq 1$,$n=1,2,3$ ,$\ldots$,(40)

then $f(z)\in K(n)$ . The equality in (40) is attained by the function$g_{2}(z)=z$ $+ \epsilon(n+1)\sum_{k=n+1}^{\infty}\frac{z^{k}}{k^{3}(k+1)}$$\epsilon=const$,$|\epsilon|=1$,$z$$\in U$.

(41)

For the generalized fractional integrals (13) we obtain then the following sufficientconditions.
Theorem 5.

Under the condition (12), if the function $f(z)$ defined by (1) satisfies

23$\sum_{k=n+1}^{\infty}k$then$\tilde{I_{(\beta_{i}),m}}^{(\gamma_{i}),(\delta_{i})}f(z)$$|a_{k}| \leq\frac{1}{\theta(n+1)}$(for $\theta(n+1)$ see (32))(42)

belongs to the class $S^{*}(n)$ .

We use again the inequality (37’), $0<\theta(k)\leq\theta(n+1)$ , valid for eachwith coefficients$k\geq n+1$ and each $n\in N$ . Then, for the function$Proof$.

$\tilde{I}f(z)=z- l-\sum_{k=n+1}^{\infty}b_{k}z^{k}$$b_{k}=\theta(k)a_{k}$, we obtain$\sum_{k=n+1}^{\infty}kb_{k}\leq\theta(n+1)\sum_{k=n+1}^{\infty}ka_{k}\leq 1$.

$\blacksquare$

Analogously, using Lemma 4, we obtain
Under the condition (12), if the function $f(z)$ defined by (1) satisfies

Theorem 6.

$\sum_{k=n+1}^{\infty}k^{2}|a_{k}|\leq\frac{1}{\theta(n+1)}$

then$\tilde{I_{(\beta_{i}),m}}^{(\gamma_{i}),(\delta.\cdot)}f(z)$

(43)

,

belongs to the class $K(n)$ .

Remark. Examples of functions satisfying conditions (42),(43) are the following functions,
and$g_{3}(z)=z+ \frac{1}{\theta(k_{0})}\frac{z^{k_{0}}}{k_{0}}$$g_{4}(z)=z+ \frac{1}{\theta(k_{0})}\frac{z^{k_{0}}}{k_{0}^{2}}$respectively, with some $k_{0}>n+1$ .
Next, to obtain another kind of characterization theorems, we use the following resultof Rusheweyh and Sheil-Small [20].

Lemma 5. Let $h(z)$ and $f(z)$ be analytic in$h(0)=f(0)=0$ ,$U$

and satisfy the condition:$h(z)* \{\frac{1+\rho\sigma z}{1-\sigma z}f(z)\}\neq 0$( $z\in U$ A{0})(44)for sorne , $\sigma\in C(|\rho|=1, |\sigma|=1)with*denoting$ the Hadamard product. Then, forfunction $F(z)$ analytic in and satisfying$a$$\rho$$U$$\Re\{F(z)\}>0$$(z\in U)$,

the inequality$\Re\{\frac{(h*Ff)(z)}{(h*f)(z)}\}>0$$(z\in U)$(45)

follows.
Now we state some characterization theorems in terms of the Hadamard product.
Theorem 7. Let us assume condition (12) and let the function $f(z)$ defined by (1)belong toand satisfy$S^{*}(n)$$h(z)* \{\frac{1+\rho\sigma z}{1-\sigma z}f(z)\}\neq 0$$(z\in U\backslash \{0\})$

(46)

24
for some , $\sigma\in C(|\rho|=1, |\sigma|=1)$ and for the function $h(z)$ defined by (19). Then,$\rho$$\tilde{I_{(\beta.),m}}^{(\gamma_{\dot{1}}),(\delta_{i})}f(z)$also belongs to $S^{*}(n)$ , i.e. under such conditions the generalized fractional

integral preserves the class$Pr$$S^{*}(n)$.

. . By Theorem 2,$o\mathrm{o}f$$\hat{I}_{(\beta_{}),m}^{1^{\gamma i}),(\delta.)}.f(z)=z+\sum_{k=n+1}^{\infty}\theta(k)z^{k}=h(z)$$*f(z)$.

Since it is easy to check that$\frac{z(h*f)’(z)}{(h*f)(z)}=\frac{(h*(zf’))}{(h*f)(z)}$,for each$h,f\in \mathrm{K}(\mathrm{n})$,it follows, if we set $F(z)=zf^{J}(z)/f(z)$ ,$\frac{z(\tilde{I}f(z))’}{(\tilde{I}f(z)}=\frac{h*zf^{l}}{h*f}=\frac{h*Ff}{h*f}$.

Using that $f\in S^{*}(n)$ implies $\Re\{F(z)\}>0$ , we obtain by Lemma 5$\Re\{\frac{z(\tilde{I}f(z))’}{(\tilde{I}f(z)}\}=\Re\{\frac{(h*Ff)(z)}{(h*f)(z)}\}>0$$\Rightarrow$$\tilde{I}f(z)$$\in S^{*}(n)$.
$\blacksquare$

For asubclass of the convex functions, an analogous theorem reads as follows.
Theorem 8.
Let us assume condition (12) and let the function $f(z)$ defined by (1)belong to $K(n)$ and satisfy$h(z)* \{\frac{1+\rho\sigma z}{1-\sigma z}zf^{l}(z)\}\neq 0$$(z\in U\backslash \{0\})$(47)

for some , $\sigma\in C(|\rho|=1, |\sigma|=1)$ and for the function $h(z)$ defined by (19). Then,. under such conditions the generalized fractional
also belongs to $K(n)$ ,$\rho$$i.e$$\hat{I}_{(\beta i_{),m}^{),(\delta_{i})}f(\chi)}^{1\gamma}$

integrals preserve the class $K(n)$ .

.

Note that in (47) we have $zf^{l}(z)$ instead of $f(z)$ in (46). We use the fact
wt
that $f\in K(n)\Leftrightarrow zf’\in S^{*}(n)$ and Theorem 7.
$Pr$ $oof$) Let $h(z)$ be convex and $f(z)$ be starlike
Lemma 6. (Rusheweyh and SheiUSmallin U. Then, for each function $F(z)$ analytic in and satisfying $\Re\{F(z)\}>0(z\in U)$ ,inequality$[\mathit{2}\theta]$$U$$d\iota e$$\Re\{\frac{(h*Ff)(z)}{(h*f)(z)}\}>0$$(z\in U)$(48)

holds valid.
Whence, in away similar like in Theorems 7,8 we have the following characterization
theorems for the generalized fractional integration operators (13)25Theorem 9. Let us assume conditions (12), and let the function $f(z)$ defined by (1)and its “convolution function” $h(z)$ defined by (19) belong to $K(n)$ . Then,
belong to
belongs to $S^{*}(n)$ , i.e.
$S^{*}(n)$$\tilde{I_{(\beta_{*}),m}}^{(\gamma.),(\delta_{i})}.\cdot f(z)$$f(z)\in S^{*}(n)$,$h(z)\in K(n)$$\tilde{I}_{(\beta\dot{.}),m}^{(\gamma_{\dot{1}}),(\delta_{i})}f(z)$$\Rightarrow$$\in S^{*}(n)$

.

(49)

Theorem 10. Let us assume conditions (12), and the functions $f(z)$ defined by (1)and the “convolution function” $h(z)$ defined by (19) belong to $K(n)$ . Then,
belongs to $K(n)$ , $i.e$ .
(50)).
$h(z)\in K(n)$$f(z)\in K(n)$ ,$\tilde{I}_{(\beta_{\dot{1}}),m}^{(\gamma_{*}),(\delta_{i})}\cdot f(z)$$\tilde{I_{(\beta.),m}}^{(),(\delta)}\gamma i.f(z)\in K\langle n$$\Rightarrow$”
Summarized, the above results (49) and (50), mean that if the $u_{Convolutionfi\iota nction}$(19) of generalized ffactional integration operator (13) belongs to $K(n)$ , then this operator, $K(n)$ .
preserves both classes$S^{*}(n)$$\hat{I}_{(\beta),m}^{(\gamma i),(\delta_{\dot{1}})}$

5. Special cases, $i=1$ , , , we obtain the analogues of TheObviously, putting in results here
orems 1- 10 for the generalized fractional integration operators with -function kernels,
see Kiryakova, Saigo and Owa [9].
Then, same type results follow for anumber of integral (or, integr0-differential anddifferential operators, when considering the respective generalized fractional derivatives)operators that are rather popular in univalent function theory but follow as special cases(mainly for $m=1,2$ and one example for $m=3$ ).
In Saigo [21],[23], the following operators of generalized fractional integration and differentiation that involve the Gauss hypergeometric function have been introduced:$\beta_{i}=1$$\ldots$$m$$G$$I^{\alpha,\beta,\eta}f(z)=z^{-\alpha-\beta} \int_{0}^{\epsilon}\frac{(z-\xi)^{\alpha-1}}{\Gamma(\alpha)}2F1(\alpha+\beta, -\eta;\alpha;1-\frac{\xi}{z})f(\xi)d\xi$

,

(51)

, . First, operator (51) has been considered for real-value
for real parametersfunctions and used for solving boundary value problems [22],[31] for the Euler-Darboux
equation, but recently Srivastava, Saigo and Owa (see for example, [32], [12]) have applied
it to classes of univalent functions.
The “normalized” operator (51) falls in the scheme of operators (13) with $m=2$ ,namely:$\alpha>0,\beta$$\mathrm{d}$$\eta$$\tilde{I}^{\alpha\beta,\eta}:=\frac{\Gamma(2-\beta)\Gamma(2+\alpha+\eta)}{\Gamma(2-\beta+\eta)}z^{\beta}I^{\alpha,\beta,\eta}=\frac{\Gamma(2-\beta)\Gamma(2+\alpha+\eta)}{\Gamma(2-\beta+\eta)}I_{(1,1),2}^{(\eta-\beta,0),(-\eta,\alpha+\eta)}(51*)$

and has respectively, multiplier sequence and convolution function of the form$\theta(k)=\frac{(-\beta+\eta+2)_{k-1}k!}{(-\beta+2)_{k-1}(\alpha+\eta+2)_{k-1}}$,

26$-\beta+\eta+2+n,$ $2+n$$\alpha+\eta+2+n1,$ ,$h(z)=z+ \frac{(-\beta+\eta+2)_{n}(n+1)!}{(-\beta+2)_{n}(\alpha+\eta+2)_{n}}z_{3}^{n+1}F_{2}(-\beta+2+n,$z).

Especially in the class $A=A(1)$ , its convolutional representation turns into:$\tilde{I}^{\alpha,\beta,\eta}f(z)=h(z)*f(z)$$-\beta+\eta+2,$ $2$$\alpha+\eta+21,$ ; $z)$ .
with $h(z)=z3F_{2}(-\beta+2,$

For the corresponding results in the classes we consider, for any $n\in N$ under conditions$\beta-\eta<2$ ,, see Kiryakova, Saigo and Owa [9].
,$\alpha+\eta\geq 0$$\eta\leq 0$

In $[3],[4]$ Hohlov introduced ageneralized fractional integration operator defined bymeans of the Hadamard product with an arbitrary Gauss hypergeometric function:$\mathrm{F}(a,b, c)f(z):=z2F1(a,$b;c; $z)*f(z)$ .

(52)

This three-parameter family of operators contains as special cases most of the known linear
integral or differential operators, already used in univalent functions theory, namely: the
Biernacki operator, Rushe weyh derivative, generalized Libe $m$ operator and its inverse,
Carlson-Shaffer operator, etc. For details, see Hohlov [3], [4], Kiryakova [7], Kiryakova et
al. [10], [11] .
This rather general Hohlov operator (52) also follows as aparticular case of generalized$)$

fractional integrals (13):$\mathrm{F}(a,$$b,c \rangle f(z)=\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}I_{(1,1),2}^{(a-2,b-2),(1-a,\mathrm{c}-b)}f(z)=\tilde{I}_{(1,1),2}^{(a-2,b-2),(1-a,c-b)}f(z)$.

$(52*)$

Thus, Theorems 1-10 give corresponding results for this operator, and also for all its
special cases. The conditions (12) now are: $0<a\leq 1,0<b\leq c$ . We will refer here only
to the form of its multipliers and convolution function, namely:$\theta(k)=\frac{(a)_{k-1}(b)_{k-1}}{(1)_{k-1}(c)_{k-1}}$

and$\Psi(n \% 1)=\frac{(a)_{n}(b)_{n}}{n!(c)_{n}}$,$h(z)=z+ \frac{(a)_{n}(b)_{n}}{n!(c)_{n}}z^{n+1}\epsilon F_{2}(1, a+n, b+n;c+n, 1+n;z)$

in $A(n)$ ,resp. in the class $A=A(1)$ : $h(z)=z2F1(a, b;c;z)$ , aresult that conforms with original
Hohlov’s representation (52).

In [24],[25] Saigo and his -worker investigated in details the operator of generalized
fractional integration which involves the s0-called Appell’s function:$\mathrm{c}\mathrm{o}$$F_{3}$$I( \alpha, d,\beta,\beta’;\gamma)f(z)=z^{-\alpha}\int_{0}^{z}\frac{(z-\xi)^{\gamma-1}}{\Gamma(\gamma)}\xi^{-\alpha’}F_{3}(\alpha,$$\alpha’,\beta,\beta’;\gamma;1-\frac{\xi}{z}$,$1- \frac{z}{\xi}f(\xi)d\xi$, (53)

but can be decomposed also as products of three Erd\’elyi-Kober operators (9). As shown by
Kiryakova [7], this is an example of generalized fractional integrals (8), (13) of multiplicity$m=3$ , and could be represented also in the form

27

$I( \alpha, \alpha’,\beta, \beta’;\gamma)f(z)=z^{-\alpha-\alpha’+\gamma}\int_{0}^{1}G_{3,3}^{3,0}[\sigma|\alpha-\alpha’+,\beta,\gamma-2\alpha’,\gamma-\alpha’-\beta’\alpha-\alpha,\beta-\alpha’,\gamma-2\alpha’-\beta’]f(z\sigma)d\sigma$

.

Then,
$I(\alpha, d,\beta,\beta’;\gamma)f(z)=z^{-\alpha-\alpha’+\gamma}I_{(1,1,1),\theta}^{(\alpha-\alpha’,\beta-\alpha’,\gamma-2\alpha’-\beta’),(\beta,\gamma-\alpha’-\beta,\alpha’)}f(z)$and for the “normalized”$F_{3}$.

$(53*)$-operator of form (13):$\tilde{I}f(z)=\tilde{I}(\alpha, \alpha’,\beta,\beta’;\gamma)f(z):=z^{\alpha+\alpha’-\gamma}I(\alpha,\alpha’,\beta, \beta’;\gamma)f(z)$we can apply all the results for classes of univalent functions, already obtained in Theorems
1–10. Let us mention that in this case the convolution function $h(z)$ expresses in terms-function. Details can be seen in Kiryakova, Saigo and Owa [9].
of the$4F_{3}$

Now, we consider some two examples of integral operators, studied recently in classesof univalent functions, that fall essentially in the case of generalized fractional integration
operators with $\beta\neq(1, 1, \ldots, 1)$ .

These are integral operators, considered in several modified forms by Raina et al.
(Raina [15], Raina end Bolia [16], Raina, Saigo and Choi [18], Raina and Kalia [17]), andothers.
The first operator, in the case of functions $f(z)$ of the class $A(n)$ , is (see for example[17, p.337, (2.3),(2.5)], and note that the $\beta>0$ here was denoted by $m$ in the original
papers by Raina et al.):$T_{C}^{A}(a, c;n)f(z)=\Phi_{C}^{A}(a, c;n;z)*f(z)$

(54)

,

with
$\Phi_{C}^{A}(a,c;n;z)=\frac{\Gamma(c+(p-1)C)}{\Gamma(a+(p-1)A)}z^{n}\sum_{k=0}^{\infty}\frac{\Gamma(a+(p-1)A+nA)}{\Gamma(c+(p-1)C+nC)}z^{k}$$= \frac{\Gamma(c+(p-1)C)}{\Gamma(a+(p-1)A)}z_{2}^{n}\Psi_{1}$($(1, 1),(a-A+nA,A)(c-C+nC, C)$

;

$z$

)

(55)

and the second, is a composition of two operators as above, $T_{C}^{A}(a, c):=T_{C}^{A}(a, c;1)(n=1$
is taken for simplicity):$M_{z_{j\beta}}^{\lambda,\mu,\eta}f(z):=T_{\beta}^{\beta}(1+\beta, 1-\mu+\beta)T_{\beta}^{\beta}(1+\eta-\mu+\beta, 1+\eta-\lambda+\beta)f(z)$.

$= \frac{\Gamma(1-\mu+\beta)\Gamma(1+\eta-\lambda+\beta)}{\Gamma(1+\beta)\Gamma(1+\eta-\mu+\beta)}z^{\mu/\beta}D_{0,.\cdot 1/\beta}^{\lambda,\mu,\eta},,f(z)$(56)

Here, for $0\leq\lambda<1;\mu,\eta\in R$, $\beta>0,\beta>\max\{\lambda-\eta-1,\mu-1\}$ ,$D_{0,z\beta}^{\lambda,\mu,\eta}f(z)= \frac{d}{dz^{\beta}}\{\frac{z^{-\beta(\mu-\lambda)}}{\Gamma(1-\lambda)}\int_{0}^{z}(z^{\beta}-t^{\beta})_{2}^{-\lambda}F_{1}(\mu-\lambda, 1-\eta;1-\lambda;1-\frac{t^{\beta}}{z^{\beta}})f(t)dt^{\beta}\}_{\mathrm{r}\mathrm{w}\backslash }$,28
is the fractional differential operator, corresponding to the s0-called modified Saigo oper),(see same papers by Raina et al., and compare with expressions in (51),ator$(51^{*})$$I_{0,Zj\beta}^{\lambda,\mu,\eta}$$I_{0_{\sim j}\beta}^{\lambda,\mu,\eta},f \vee(z)=\frac{z^{-\beta(\lambda+\mu)}}{\Gamma(\lambda)}\int_{0}^{z}(z^{\beta}-t^{\beta})_{2}^{\lambda-1}F_{1}$(X%$- \eta;\lambda;1-\frac{t^{\beta}}{z^{\beta}}$$\mu,$)$f(t)dt^{\beta}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$.

$\tilde{I_{(\beta,\beta),2}}^{\lambda,\mu,\eta}f(z)$,(58)
in our denotations (8),(13).
It is seen then, that for $\beta=1$ , $n=1$ , $A=C=1,0<a<c$, the operator
reduces to the Carlson-Shaffer integral operator $L(a,c)$ , defined by aHadamard productwith aGauss function, and easily seen to be special case of the Erd\’elyi-Kober operators(9) (see e.g. [9]):$T_{C}^{A}(a, c;n)$(59)$\mathrm{L}(a, c)f(z)=\Phi(a, c;z)*f(z)=\{z_{\mathit{2}}F_{1}(1, a;c;z)\}*f(z)$$= \frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)}\int_{0}^{1}(1-\sigma)^{\mathrm{c}-a-1}\sigma^{a-2}f(z\sigma)d\sigma=\frac{\Gamma(c)}{\Gamma(a)}\Gamma_{1}^{-2,\mathrm{c}-a}f(z)$.

, reduce for $\beta=1$ to the hypergeometricor
The operators (56),(57), i.e.
fractional derivative (resp. integral (51) with aGauss function, studied by Saigo et
The operators (54)-(55) with $A=C=1/\beta$ and (56)-(57)-(58) are special cases of thegeneralized fractional integrals (8),(13), resp. for $m=1$ and $m=2$ (with $A_{1}=C_{1}=$$A_{2}=C_{2}=1/\beta$ , i.e. $\beta_{1}=\beta_{2}=\beta>0$ ). Evidently, $m$ -tuple compositions of operators(54)-(55) give operators of form (13) in the general case $m>1$ .
Results for above two operators have been obtained by Raina et al., for example as), and respective operational properties of both
follows: analogue of Lemma 0(for,operators-in Raina [15], where as applications some inequalities for the, (4) have been derived; results analogous to our characterization
Wright functions
theorems (Theorems 7-8, 9-10)-for.in Raina, Saigo and Choi [18], and-forin Raina and Kalia [17], etc. Evidently, the results presented here for generalized fractional
calculus operators (13) give, as special cases, also aseries of other corresponding analogues
for the mentioned two operators.
$M_{z\beta}^{\lambda,\mu,\eta}$$D_{0,z\beta}^{\lambda,\mu,\eta}$$al$$D_{0,\acute{z}\beta}^{\lambda\mu,\eta}$$D_{0,z\beta}^{\lambda,\mu,\eta}$$M_{z\beta}^{\lambda,\mu,\eta}$$\mathrm{p}q\Psi$$M_{z\beta}^{\lambda,\mu,\eta}$$D_{0,z\beta}^{\lambda\mu,\eta}$

Acknowledgements
by Bulgarian
The present work is partly supported by Grant MM 1305/2003
Ministry of Education and Science of Bulgaria, NSF and by Science Promotion Fund fromthe Japan Private School Promotion Foundation.
$(” \mathrm{F}\mathrm{C}\mathrm{A}\mathrm{A}")$*References[1] S. K. Chatterjea: On starlike functions, J. Pure Math. $1(1981)$ , 23-26.

[2] A. Erd&yi, W. Magnus, F. Oberhettinger and
1-3, McGraw-Hill, New York, 1953

. Soni: Higher$\mathrm{R}.\mathrm{P}$$[]$}$\cdot anscendentd$ functions Vol-s

29[3] Yu. Hohlov: Convolutional operators preserving univalent functions (In Russian), Ukrain. Mat .
37(1985), 220-226.

$J$.

[4] Yu. Hohlov: Convolutional operators preserving univalent functions, Pliska. Studia Math. Bulg.
10(1989), 87-92.
, Jae Ho Choi, M. Saigo: Some properties of convolution opeartors involving carlsonshaffer operator, Panamerical Math. J. 102000, No 2, 51-60.

[5] Yong Chan$\mathrm{k}\mathrm{i}\mathrm{m}$-function fractional integration operators in some classes of ana[6] V. Kiryakova: Generalized
lytic functions, Bull. Mathematique (Beograd) 40(1988), 259-266.
$H_{m,m}^{m,0}$[7] V. Kiryakova: Generalized Fractional Calculus and Applications (Pitman
301), Longman, Harlow (UK), 1994.

${\rm Res}$. Notes in Mat. Series,[8] V. Kiryakova: Multiple (multiindex) Mittag-leffler functions and relation sto generalized fractional
calculus, J. Comput A $ppl$ . Math. 118, 241-259, 2000.

[9] V. Kiryakova, M. Saigo, S. Owa: Distortion and characterization theorems for starlike and convex
functions related to generalized fractional calculus, In: “Kokyuroku” ( . Inst. Math. Sci., Kyoto
Univ.), 245-46, 1997.
${\rm Res}$. Srivastava: Some criteria for univalence of analytic functions in[10] V. Kiryakova, M. Saigo,
volving generalized fractional calculus operators, $Ihct:onal$ Calculus &Applied Analysis 1, No 1
(1998), 79-104.
$\mathrm{H}.\mathrm{M}$. Srivastava: Generalized multiple Riemann-Liouville ffactional differintegrals and their applications in univalent function theory, Analysis, Geometry and Groups:
Riemann Legacy Volume, Hadronic Press, Florida, 191-226, 1993.

[11] V. Kiryakova and$\mathrm{H}.\mathrm{M}$$A$. Srivastava: Some characterization theorems for starlike and convex[12] S. Owa, M. Saigo andfunctions involving acertain fractional integral operator, J. Math. Anal, and Appl. 140(1989),$\mathrm{H}.\mathrm{M}$419-426.
[13] A. Prudnikov, Yu. Brychkov and. Marichev: Integrals and Series. Vol. :Some More Special
Functions, Gordon&Breach Sci. Publ., Switzerland, 1993.
$S$$\mathrm{O}.\mathrm{I}$. Marichev: IntegralS and Series. Vol. :Some More Special[14] A. Prudnikov, Yu. Brychkov andSwitzerland, 1993.
Publ.,
Functions, Gordon&Breach Sci.
$\theta$$\mathrm{O}.\mathrm{I}$[15] R. K. Raina: On certain classes of analytic functions and applications to fractional calculus oPeraotrs, Integral Transforms Spec. hnct. (1997), 247-260.
$\epsilon$[16] R. K. Raina, M. Bolia: On distortion theorems involving generalized fractional calculus operators,
Tamkang J. Mathematics 27 (1996), No 3, 233-241.

[17] R. K. Raina, R. N. Kalia: Characterizations for subclasses of analytic functions conecting linear, No 4, 335350.
. Calc. &Appl. Anal,ffactioml calculus operators,$fi\vdash act$

[18]$\mathrm{R}.\mathrm{K}$[19]$\mathrm{M}.\mathrm{S}$$\mathrm{b}\mathrm{f}1(1998)$. Raina, M. Saigo, Jae ho Choi: Anote on acertain linear operator and its applications tocertain subclasses of analytic functions, J. fihct. Calculus 14, November (1998), 45-51.

. Robertson: On the theory of univalent functions, Ann. of Math. 37(1936), 374-408.

[20] S. Rusheweyh, T. Sheil-Small: Hadamard products of schlicht functions and the Polya-Schoenberg
conjecture, Corn rnent. Math. Helv. 48(1973), 119-135

30[21] M. Saigo: Aremark on integral operators involving the Gauss hypergeometric functions, Math.
Rep. College General Ed. Kyushu Univ. 11(1978), 135-143.

[22] M. Saigo: Acertain boundary value problem for the Euler-Darboux equation, Math. Japon.
24(1979), 377-385; . ibid. 25(1980), 211-220; III, ibid. 26(1981), 103-119.
$\mathrm{I}\mathrm{I}$[23] M. Saigo: Ageneralization of fractional calculus, Fractional Calculus (Proc. Workshop held at Ross
Priory, Univ. Strathclyde), Pitman, 188-198, 1985.
[24] M. Saigo: On generalized ffactional calculus operators. Recent Advances in Appl. Mathematics(Proc. Int. Workshop Kuwait Univ.), 441-450, 1996.
[25] M. Saigo and N. Maeda: More Generalization of Fractional Calculus, Transfo $m$ Methods &Special
Functions, Varna’96 (Proc. 2nd Int. Workshop), SCTP, 386-400, 1997.
[26] Samko, S.G., Kilbas, A.A., Marichev,:fihctional Integrals and Derivatives (Theory and Applications). Gordon&Breach Sci. Publ., N. York etc. (1993).
$\mathrm{O}.\mathrm{I}.$[27] H. Silverman: Univalent functions with negative coefficients, Proc. $AMS51(1975)$ , 109116.
[28] Srivastava, H.M., Gupta, K.C., Goyal,: The $H$ -fihnctions of One and Teoo Variables with
Applications. South Asian Publ., New Delhi (1982).
$\mathrm{S}.\mathrm{P}.$. Srivastava and S. Owa: An application of the fractional derivative, Math. Japon. 29(1984),
383-389.

[29]$\mathrm{H}.\mathrm{M}$[30]$\mathrm{H}.\mathrm{M}$

$Sem$

[31]
[32]

. Srivastava, S. Owa and$\mathrm{S}.\mathrm{K}$. Chatterjea: Anote on certain classes of starlike functions, Rend.

. Mat. Univ. Padova 77(1987), 115-124.

. Srivastava and M. Saigo: Multiplication of fractional calculus operators and boundary value
problems involving the Euler-Darboux equation, J. Math. Anal. Appl. 121(1987), 325-369.
$\mathrm{H}.\mathrm{M}$$\mathrm{H}.\mathrm{M}$. Srivastava, M. Saigo and S. Owa: Aclass of distortion theorems involving certain operatorsof fractional calculus of starlike functions, J. Math. Anal and Appl. 131(1988), 412-420