数理解析研究所講究録 1341 巻 2003 年 85-93

85

Inequalities for Saigo’s Fractional Calculus Operator
Shigeyoshi Owa 1

Megumi Saigo 2

Virginia S. Kiryakova 3

Abstract
which
Let be the class of functions $f(z)$ of the form $f(z)=z$$\inA$, the subclassof satisfying
are analytic in the open unit disk . For $f(z)$$(k \geq 2)$ is introduced. The object of the present
the coefficient inequalitiesofpaper is to derive some inequalities for Saigo’s fractional calculus operator$+a_{2}z^{2}\star a_{3}z^{3}+\cdots$

$A$

$A$

$A(n,\delta)$$U$$|a_{k}|\leq k^{n+\delta}$$f(z)\in A(n,\delta)$$I_{0,*}^{a,\beta,\eta}f(z)$.

2000 Mathematics Subject Classifications: Primary$30\mathrm{C}45$, Secondary$26\mathrm{A}33$

Key words and phrases: analytic function, Saigo’s fractional calculus operator, Owa’s
fractional calculus operator
1. Introduction

Let A be the class of functions $f(z)$ of the form
(1.1)

$f(z)=z+ \sum_{k=2}^{\infty}a_{k}z^{k}$which are analytic in the unit disk $U=\{z \in C : |z|<1\}$ . Saigo’s fractional calculus
operatorof $f(z)\in A$ is defined in Srivastava, Saigo and Owa [7] (see also Saigo[5] as follows.
$I_{0,s}^{a,\beta,\eta}f(z)$$)$

For real numbers $\alpha>0,\beta$ and , the fractional integral operator
of $f(z)$ is defined by

Definition 1.1.
$I_{0_{1}z}^{\alpha\beta,\eta}f(z)$$\eta$$-\eta$$\Gamma_{0,z}^{\beta_{\mathrm{I}}\eta}’ f(z)=\frac{z^{-\alpha-\beta}}{\Gamma(\alpha)}\int_{0}^{z}(z-\zeta)_{2}^{\alpha-1}F_{1}(\alpha+\beta\alpha’$;$1- \frac{\zeta}{z})f(\zeta)d\zeta$,

1 Department of Mathematics, Kinki University, Higashi-Osaka, Osaka 577, Japan

2 Department of Applied Mathematics, Fukuoka University, Fukuoka 814–0180Japan

3 Institute of Mathematics and Informatics, Bulgarian Academ of Sciences, Sofia 1090, Bulgari$\mathrm{y}$

(1.2)

86
where $f(z)$ is an analytic function in asimply-connected region of the -plane containing
the origin with the order$(zarrow 0)$ ,$f(z)=O(|z|^{c})$
$z$

where
$\epsilon$

and the multiplicity of$z-\zeta>0$$(z-\zeta)^{\alpha-1}$.

Remark 1.

$> \max\{0,\beta-\eta\}-1$

is removed by requiring $\log(z-\zeta)$ to be real when

It follows from Definition 1.1 that
$I_{0,z}^{\alpha,-\alpha,\eta}f(z)=D_{l}^{-\alpha}f(z)$

when

,

$\beta=-\alpha$, where$D_{z}^{-\alpha}$$= \int_{0}^{z}\frac{f(\zeta)}{(z-\zeta)^{1-\alpha}}d\zeta$

,

(1.3)

is the fractional integral of order adefined by Owa [3].

Definition 1.2. For real numbers $0\leq\alpha<1,\beta$ and , the fractional derivative operator
of $f(z)$ is defined by$\eta$$J_{0,z}^{\alpha,\beta,\eta}f(z)$$J_{0,z}^{\alpha,\beta,\eta}f(z)$$1-\eta 1-\alpha$$= \frac{1}{\Gamma(1-\alpha)}\frac{d}{dz}\{z^{\alpha-\beta}\int_{0}^{z}(z-\zeta)^{-\alpha_{2}}F_{1}(\beta-\alpha,$; $1- \frac{\zeta}{z})f(\zeta)d\zeta\}(1.3)$

and$J_{0,s}^{m+\alpha,\beta,\eta}f(z)= \frac{d^{m}}{dz^{m}}(J_{0,z}^{\alpha fl}"’ f(z))$$(m=0,1,2, \ldots)$

,

(1.5)

where $f(z)$ is an analytic function in asimply-connected region of the -plane containing
the origin with the order$(zarrow 0)$ ,$f(z)=O(|z|^{e})$
$z$

where
$\epsilon$

and the multiplicity of$(z-\zeta)^{-\alpha}$$> \max\{0,\beta-\eta\}-1$

is removed as in Definition 1.1 above.

Remark 2. We also note that, when
$J_{0,z}^{\alpha,\alpha,\eta}f(z)$,$\beta=\alpha$,$=D_{z}^{\alpha}f(z)= \frac{1d}{\Gamma(1-\alpha)dz}\{\int_{0}^{z}\frac{f(z)d\zeta}{(z-\zeta)^{\alpha}}\}$

(1.6)

and
$\sqrt{0}^{+\alpha,\alpha,\eta},f’(z)=D_{z}^{m+\alpha}f(z)$$= \frac{ff^{n}}{dz^{m}}(D_{z}^{\alpha}f(z))$

,

(1.7)

where
and
are fractional derivatives of $f(z)$ defined by Owa [3].
Afunction $f(z)\in A$ is said to be starlike in if it satisfies$D_{z}^{\alpha}f(z)$$D_{z}^{m+\alpha}f(z)$$U$$\Re(\frac{zf’(z)}{f(z)})>0$(a $\in U$ ).

(1.8

87 87
We denote by
the subclass of A consisting of all starlike functions in U. Afunction$f(z)\in A$ is said to be convex in U if it satisfies$S^{*}$$\Re(1+\frac{zf’(z)}{f’(z)})>0$(z $\in U)$ .

(1.9)

We also denote by K the subclass of A consisting of functions $f(z)$ which are convex.
in U. Note that $f(z)\in K$ if and only if $zf^{J}(z)$$\in S^{*}$

It is well-known that:(i) if $f(z)$$\in S^{*}$, then $|a_{k}|\leq k(k =2,3,4,$\ldots ), see e.g. [1],and(ii) if $f(z)\in K$ , then $|a_{k}|\leq 1(k=2,3,4, \ldots)$ , see e.g. [4].

Now, let $A(n,\delta)$ be the subclass of$A$consisting of all functions $f(z)$ which satisfy(1.10)$|a_{k}|\leq k^{n+\delta}$

for some $n=0,1,2$ ,$K\subset A(0,0)$.

$\ldots$, and for some $0\leq\delta\leq 1$ . Then we see that $S^{*}\subset A(1,0)$ and

2. InequalitiesLet
be the generalized hypergeometric function defined by (for all details, see[2]$\mathrm{p}qF(z)$$)$$pqF(z)\equiv F(pq\alpha_{1},\alpha_{2},\ldots,\alpha_{p}\beta_{1},\beta_{2},\ldots,\beta_{q}$where$(\alpha_{j})_{k}$;$z)= \sum_{k-\mathrm{l}}^{\infty}(\frac{\prod_{j=1}^{\mathrm{p}}(\alpha_{j})_{k}}{j=1\mathrm{f}\mathrm{i}(\beta_{\mathrm{j}})_{k}})\frac{z^{k}}{(1)_{k}}$

,

(2.1)

means the Pochhammer symbol defined by$(\alpha_{j})_{k}=\{$

1$(k=0)$$\alpha_{j}(\alpha_{j}+1)\ldots$$(\alpha_{j}+k-1)$$(k=1,2,3, \ldots)$

.

(2.2)

In order to derive our inequalities for Saigo’s fractional calculus operators, we need thefolowing lemma due to Srivastava, Saigo and Owa [7).
and be real. Then, for $k> \max\{0,\beta-\eta\}-1$ ,Lemma 2.1. Let$\alpha>0,\beta$$\eta$$I_{0,\acute{z}}^{\alpha\beta,\eta}z^{\beta}= \frac{\Gamma(k+1)\Gamma(k-\beta+\eta+1)}{\Gamma(k-\beta+1)\Gamma(k+\alpha+\eta+1)}z^{k-\beta}$

We also have
Lemma 2.2. let $0\leq\alpha<1,\beta$ and$\eta$.

(2.3)

be real. Then, for $k> \max\{0,\beta-\eta\}-1$ ,$J_{0,z}^{\alpha fl\pi}’ z^{k}= \frac{\Gamma(k+1)\Gamma(k-\beta+\eta+1)}{\Gamma(k-\beta+1)\Gamma(k-\alpha+\eta+1)}z^{k-\beta}$

(2.4)

88

$J_{0,z}^{m+\alpha,\beta,\eta}z^{k}= \frac{\Gamma(k+1)\Gamma(k-\beta+\eta+1)}{\Gamma(k-\beta-m+1)\Gamma(k-\alpha+\eta+1)}z^{k-\beta-m}$

Our first inequality for
Theorem 2.1.
$2+\alpha+\eta>0$$I_{0,\acute{z}}^{\alpha\beta,\eta}f(z)$(m $=0,$ 1,2, \ldots ).

(2.5)

is contained in the following theorem.

Let $\alpha>0,\beta$ and be real, and let $2-\beta>0,2-\beta+\eta>0$ , and, then$\eta$. If$f(z)\in A(n,\delta)$$|I_{0,z}^{\alpha\beta,\eta}f(z)| \leq\frac{\Gamma(2-\beta+\eta)}{\Gamma(2-\beta)\Gamma(2+\alpha+\eta)}|z|_{n+8}^{1-\beta}F_{n+2}(1,$$\ldots 2,,\cdots,2,2-\beta+\eta 1,2-\beta,2+\alpha+\mathrm{y}7$;$\mathrm{E}1)$(2.6)

for $0<|z|<1$ .
Proof. Applying Lemma 2.1 for $f(z)\in A(n$ , !), we have$|I_{0,\acute{z}}^{\alpha\beta,\eta}f(z)|=| \sum_{k=1}^{\infty}\frac{\Gamma(k+1)\Gamma(k-\beta+\eta+1)}{\Gamma(k-\beta+1)\Gamma(k+\alpha+\eta+1)}a_{k}z^{k-\beta}|$$(a_{1}=1)$(2.7)$\leq\sum_{k=1}^{\infty}\frac{\Gamma(k+1)\Gamma(k-\beta+\eta+1)}{\Gamma(k-\beta+1)\Gamma(k+\alpha+\eta+1)}k^{rl+1}|z|^{k-\beta}$$= \sum_{k=0}^{\infty}\frac{\Gamma(k+2)\Gamma(k-\beta+\eta+2)}{\Gamma(k-\beta+2)\Gamma(k+\alpha+\eta+2)}(k+1)^{n+1}|z|^{k+1-\beta}$

.

Since
$\Gamma(k+\gamma)=\Gamma(\gamma)(\gamma)_{k}$$(\gamma>0)$

and$k+1= \frac{(2)_{k}}{(1)_{k}}$,we obtain that$|I_{0,z}^{\alpha,\beta,\eta}f(z)| \leq\frac{\Gamma(2-\beta+\eta)}{\Gamma(2-\beta)\Gamma(2+\alpha+\eta)}|z|^{1-\beta}(\sum_{k\#}^{\infty}\frac{((2)_{k})^{n+2}(2-\beta+\eta)_{k}}{((1)_{k})^{n}(2-\beta)_{k}(2+\alpha+\eta)_{k}}\frac{|z|^{k}}{(1)_{k}})$$= \frac{\Gamma(2-\beta+\eta)}{\Gamma(2-\beta)\Gamma(2+\alpha+\eta)}|z|_{n+\}^{1-\beta}F_{n+2}(1,$ $\ldots 2,,\cdots,2,$$2-\beta+\eta 1,2-\beta,2+\alpha+\eta$$(2.8)$; $|z|)$

which completes the proof of the theorem.
If we takein Theorem 2.1, then we have$\blacksquare$$\beta=-\alpha$Corollary 2-1.

If $f(z)$$\in A(n,\delta)$, then$|D_{l}^{-\alpha}f(z)| \leq\frac{1}{\Gamma(2+\alpha)}|z|_{n+2}^{1+\alpha}F_{n+1}(1,$for $0<|z|<1$ and$\alpha>0$\ldots 2, \c1,dots$,2$$2+\alpha$; $|z|)$(2.9).

Taking special values of n and$\delta$in Theorem 2.1, we derive the following corollary

89
Corollary 2.2.
$2+\alpha+\eta>0$. If

Let $\alpha>0,\beta$ and$\eta$

be real, and let $2-\beta>0,2-\beta+\eta>0$ , and$f(z)\in A(1,0)$ , then
$2-\beta+\eta$; $|z|)$(2.10); $|z|)$(2.12)$|I_{0,z}^{\alpha,\beta,\eta}f(z)| \leq\frac{\Gamma(2-\beta+\eta)}{\Gamma(2-\beta)\Gamma(2+\alpha+\eta)}|z|_{4}^{1-\beta}F_{8}(1,2-\beta,2+\alpha+\eta 2,2,2,$

for $0<|z|<1$ . If $f(z)\in A(0,0)$ , then$|I_{0,z}^{\alpha,\beta,\eta}f(z)| \leq\frac{\Gamma(2-\beta+\eta)}{\Gamma(2-\beta)\Gamma(2+\alpha+\eta)}|z|_{3}^{1-\beta}F_{2}(_{2-\beta,2+\alpha+\eta}2,2,2-\beta+\eta$

SimUarly, for Saigo’s fractional derivative operator$J_{0,z}^{\alpha,\beta,\eta}f(z)$of $f(z)$ , we have

Let $0\leq\alpha<1,\beta$ and be real, and let $2-\beta-m>0,2-\beta+\eta>0$,$2-\alpha+\eta>0_{f}$ and $m=0,1,2$ ,If $f(z)\in A(n,\delta)$ , thenTheorem 2.2.

$\eta$$\ldots$$|J_{0,z}^{m+\alpha,\beta,\eta}f(z)| \leq\frac{\Gamma(2-\beta+\eta)}{\Gamma(2-\beta-m)\Gamma(2-\alpha+\eta)}$$\mathrm{x}|z|_{n+3}^{1-\beta-m}F_{n+2}(1,$$\ldots$

for $0<|z|<1$ .
Proof.

2, ..., 2, $2-\beta+\eta$; $|z|)$, 1, $2-\beta-m,2-\alpha+\eta$(2.12)

Since Lemma 2.2 implies that
$J_{0,z}^{m+\alpha,\beta,\eta}f(z)$$= \sum_{k=1}^{\infty}\frac{\Gamma(k+1)\Gamma(k-\beta+\eta+1)}{\Gamma(k-\beta-m+1)\Gamma(k-\alpha+\eta+1)}a_{k}z^{k-\beta-m}$,

we easily see the inequality (2.12).
If we putin Theorem 2.2, then we have

(2.13)$\blacksquare$$\beta=\alpha$Corollary 2.3.

If $f(z)\in A(n,\delta)$ ,$\#\iota en$$|D_{z}^{\alpha}f(z)| \leq\frac{1}{\Gamma(2-\alpha-m)}|z|^{1-\alpha-m_{\mathrm{B}+2}}F_{n+1}(1,$ $\ldots,2,\ldots,21,2-\alpha-m\mathrm{i}|z|)$(2.14)

for $0<|z|$ $<1,0\leq\alpha<1$ and m $=0,$ 1.
Taking special values for and , we have
Corollary 2.4. $Lei$ $0\leq\alpha<1,\beta$ and yy be real, and let $2-\beta-m>0,2-\beta+\eta>0$ ,$2-\alpha+\eta>0$ and $m=0,1,2$ ,. If $f(z)\in A(1,0)$ , then$\delta$$n$$\ldots$,$| \sqrt{0}^{+\alpha\beta,\eta},fz(z)|\leq\frac{\Gamma(2-\beta+\eta)}{\Gamma(2-\beta-m)\Gamma(2-\alpha+\eta)}|z|^{1-\beta-m_{4}}F$$(\begin{array}{lll}2,2,2,2- \beta+\eta |z|1,2-\beta-m,2- \alpha+\eta "\tau/\backslash \end{array})\mathrm{r}$90
for $0<|z|<1$ . If $f(z)$ $\in A(0,0)$ , then$|J_{0,z}^{m+\alpha,\beta,\eta}f(z)| \leq\frac{\Gamma(2-\beta+\eta)}{\Gamma(2-\beta-m)\Gamma(2-\alpha+\eta)}|z|^{1-\beta-m_{3}}F_{2}(_{2-\beta}2,$,$2-$ $2-\beta+\eta m,$$2-\alpha+\eta$; $|z|)$ .
(2.16)

3. Appendix
We introduce the following formula for the generalized hypergeometric functions ([2]).
Lemma 3.1. (Srivastava [6]) If$\mathrm{p}qF$

(

,

;

$\beta_{1}+m,\alpha\beta_{1},\beta_{2}$$2\ldots’\ldots,\beta_{q}’\alpha_{\mathrm{P}}$is a positive integer, then$m$$z)= \sum_{j=0}^{m}$$(\begin{array}{l}mj\end{array})$$(. \frac{\frac{fi}{-}2(\alpha_{s})_{j}}{\prod_{\iota=1}^{q}(\beta_{*})_{j}})z_{\mathrm{p}-1}^{\dot{\mathrm{J}}}F_{q-1}(\alpha\rho_{2}^{2}+j,\ldots,\beta_{q}+j+j,\ldots,\alpha_{p}+j$; $Z)$(3.1)

In view of Lemma 3.1, for $n+sF_{n+2}$ in Theorem 2.1, we see that$n+\^{F}n+2$(1, . , 1, 2’$2.’$$2+\alpha+\eta 2,2-\beta+\eta$$-\beta,$;$|z|)= \sum_{j=0}^{1}$$\frac{((2)_{j})^{n+1}(2-\beta+\eta)_{j}}{((1)_{j})^{n}(2-\beta)_{j}(2+\alpha+\eta)_{j}}$$(\begin{array}{l}1j\end{array})$

(3.2)
’

$\mathrm{x}|z|_{n+2}^{j}F_{n+1}(1+j,2\ldots+,j$ $1+\cdots j’,2-\beta+j,2+\alpha+\eta+j2+j,2-\beta+\eta+j$$=F_{n+1}n+2(1$

$2,$

,.

, 1,

2’

$2,2-\beta+\eta$$-\beta,2+\alpha+\eta$; $|z|)$; $|z|)$$+ \frac{2^{n+1}(2-\beta+\eta)}{(2-\beta)(2+\alpha+\eta)}|z|_{n+2}F_{n+1}(2,$ $\ldots 3,,\cdots,3,3-\beta+\eta 2,3-\beta,3+\alpha+\eta$$=F_{n}n+1(1$,.

$2-\beta+\eta 1,2-\beta,2+\alpha+\eta$$2,,\cdots,2,$; $|z|)$$+ \frac{3\cdot 2^{n}(2-\beta+\eta)}{(2-\beta)(2+\alpha+\eta)}|z|_{n+1}F_{n}(_{2,\ldots,2,3-\beta,3+\alpha+\eta}3,\ldots,3,3-\beta+\eta$$+ \frac{4\cdot 3^{n}(2-\beta+\eta)(3-\beta+\eta)}{(2-\beta)(3-\beta)(2+\alpha+\eta)(3+\alpha+\eta)}|z|_{n+1}^{2}F_{n}$$=\ldots$

Therefore, Corollary 2.2 can be written as:; $|z|)$(3,; $|z|)$$\ldots 4,,\cdots,4,4-\beta+\eta 3,4-\beta,4+\alpha+\eta$

;

$|z|$

).

91
Corollary 3.1.

Let $\alpha>0,\beta$ and$2+\alpha+\eta>0$ . If $f(z)\in A(1,$ 0), then$\eta$

be real, and let $2-\beta>0,2-\beta+\eta>0$, and$2,$; $|z|)$$|I_{0,\mathrm{z}}^{\alpha\beta,\eta}f(z)| \leq\frac{\Gamma(2-\beta+\eta)}{\Gamma(2-\beta)\Gamma(2+\alpha+\eta)}\epsilon F_{2}(2-\beta 2,$ $2-\beta+\eta 2+\alpha+\eta$,$+ \frac{4\Gamma(2-\beta+\eta)}{\Gamma(3-\beta)\Gamma(3+\alpha+\eta)}|z|_{3}F_{2}$for $0<|z|<1$ .

$h\hslash hemm$($3,3,3-\beta+\eta$, let $\alpha>0,\beta<2$ , and$\eta$

;

$|z|$

)

(3.3)

be a positive integer.

$f(z)$ $\in A(1,0)$ , then

(3.4)$|I_{0\acute{p}}^{\alpha\beta,\eta}f(z)| \leq\frac{\Gamma(2-\beta+\eta)}{\Gamma(2-\beta)\Gamma(2+\alpha+\eta)}|z|^{1-\beta}$$\mathrm{x}\{\sum_{j=0}^{\eta}$$(\begin{array}{l}\eta j\end{array})$

$\{$

If

$\frac{((2)_{j})^{2}}{(2-\beta)_{j}(2+\alpha+\eta)_{j}}2\mathrm{F}1$$(2+\eta,2+\eta 2+\alpha+2\eta$$+ \frac{((2)_{j+1})^{2}(2-\beta+\eta)}{(2-\beta)_{\mathrm{j}+1}(2+\alpha+\eta)_{j+1}}|z|_{2}F_{1}$(; $|z|)$;$3+\eta,\+\eta 3+\alpha+2\eta$

$|z|$

)

$\}|z|^{j}\}$for $0<|z|<1$ . If $f(z)\in A(0,0)$ , then
(3.5)$|I_{0,f(z)|\leq\frac{\Gamma(2-\beta+\eta)}{\Gamma(2-\beta)\Gamma(2+\alpha+\eta)}|z|^{1-\beta}}^{\alpha_{\acute{l}}\beta,\eta}$$\mathrm{x}\{\sum_{j=0}^{\eta}$$(\begin{array}{l}\eta j\end{array})$$\frac{((2)_{j})^{2}}{(2+\alpha+\eta)_{j}}|z|_{2}^{\mathrm{j}}F_{1}$($2+\eta,2+\eta 2+\alpha+2\eta$

;

$|z|$

)

$\}$

for $0<|z|<1$ .
We also see from Corollary 2.4 that
Corollary 3.2. Let $0\leq\alpha<1,\beta$ and be real, and let $2-\beta-m>0,2-\beta+\eta>0$ ,$2-\alpha+\eta>0$ and $m=0,1,2$ ,. If $f(z)\in A(1,0)$ , then$\eta$$\ldots$$|J_{0,z}^{n+\alpha,\beta,\eta}f(z)| \leq\frac{\Gamma(2-\beta+\eta)}{\Gamma(2-\beta-\sqrt)\Gamma(2-\alpha+\eta)}|z|^{1-\beta-m_{3}}F_{2}(2-\beta-2,2,$ $2-\beta+\eta m,2-\alpha+\eta$

3, 3, $3-\beta+\eta$$+ \frac{4\Gamma(3-\beta+\eta)}{\Gamma(3-\beta-m)\Gamma(3-\alpha+\eta)}|z|^{2-\beta-m_{3}}F_{2}(_{3-\beta-m,3-\alpha+\eta}$

for $0<|z|<1$ . hhhe ,$mooe$, let $0\leq\alpha<1$ andinteger, and let $2-\beta-m>0$ and $m=0,1,2$ ,$\ldots$$\beta<2$

If

be real, and$f(z)\in A(1,0)$$\eta$, then; $|z|)$; $|z|)$(3.6)
be a positive$|J_{0,z}^{m+\alpha,\beta,\eta}f(z)| \leq\frac{\Gamma(2-\beta+\eta)}{\Gamma(2-\beta-m)\Gamma(2-\alpha+\eta)}|z|^{1-\beta-m}$/0$’\backslash$92x$\{\sum_{j=0}^{\eta+m}(_{j}^{\eta+m})\{$ $\frac{((2)_{j})^{2}}{(2-\beta-m)_{\mathrm{j}}(2-\alpha+\eta)_{j}}$$+ \frac{((2)_{j+1})^{2}(2-\beta+\eta)}{(2-\beta-m)_{j+1}(2-\alpha+\eta)_{j+1}}|z|_{2}F_{1}$(2F1$(2+\eta+m,2+\eta+m2-\alpha+2\eta+m$$3+\eta+m,3+\eta+m3-\alpha+2\eta+m$;

for $0<|z|$ $<1$ . If $f(z)$ $\in A(0,0)$ , then

; $|z|)$|z|)}$|z|^{j}\}$$| \sqrt{0}^{+\alpha,\beta,\eta},,f(z)|\leq\frac{\Gamma(2-\beta+\eta)}{\Gamma(2-\beta-m)\Gamma(2-\alpha+\eta)}|z|^{1-\beta-m}$

(3.8)

x

$\{\sum_{j-\sim}^{\eta+m}$$(\begin{array}{l}\eta+mj\end{array})$$\frac{((2)_{j})^{2}}{(2-\beta-m)_{\mathrm{j}}(2-\alpha+\eta)_{j}}|z|_{2}^{j}F_{1}$($2+\eta+m,2+\eta+m2-\alpha+2\eta+m$

;

|z|)

$\}$

for $0<|z|$ $<1$ .
Further, we consider the case of f3 $=-\alpha$ in Corollary 2.2.

Corollary 3.3. Let $\alpha>0$ . If $f(z)$ $\in A(1,0)$ , then$|D_{z}^{-\alpha}f(z)| \leq\frac{|z|^{1+\alpha}}{\Gamma(2+\alpha)(1-|z|)^{S-\alpha}}$$\mathrm{x}\{(1-|z|)_{2}F_{1}(_{2+\alpha}\alpha,\alpha$;$|z|)+ \frac{4|z|}{2+\alpha}2F_{1}(_{3+\alpha}\alpha,\alpha$; $|z|)\}$(3.9)

for $z\in U$ . If $f(z)\in A(0,0)$ , then$\alpha$$|D_{l}^{-\alpha}f(z)| \leq\frac{|z|^{1+\alpha}}{\Gamma(2+\alpha)}2F1(2+\alpha\alpha,$

for

$z$

$\in U$; $|z|)$(3.10).

.

$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{f}$Applying Lemma 3.1 and the formula (see [2])$2F_{1}$($\alpha,\beta\gamma$;$z)=(1-z)_{2}^{\gamma-\alpha-\beta}F_{1}(\gamma-\alpha,\gamma-\beta\gamma$(3.11); $z)$

we see that$|D_{z}^{-\alpha}f(z)| \leq\frac{|z|^{1+\alpha}}{\Gamma(2+\alpha)}\S^{F_{2}}(1,2+\alpha 2,2,2$$= \frac{|z|^{1+\alpha}}{\Gamma(2+\alpha)}\{\sum_{\mathrm{j}=0}^{1}$$(\begin{array}{l}1j\end{array})$$\frac{(2)_{j}(2)_{\mathrm{j}}}{(1)_{j}(2+\alpha)_{j}}|z|_{2}^{j}F_{1}$((3.12); $|z|)$$2+j,$.

$2+j2+\alpha+j|$ $|z|$

)

$\}$

93

$= \frac{|z|^{1+\alpha}}{\Gamma(2+\alpha)}\{\sum_{\mathrm{j}=0}^{1}$$(\begin{array}{l}1j\end{array})$$\frac{(2)_{j}(2)_{j}}{(1)_{j}(2+\alpha)_{j}}\frac{|z|^{j}}{(1-|z|)^{2+j-\alpha}}2F1$$= \frac{|z|^{1+\alpha}}{\Gamma(2+\alpha)(1-|z|^{3-\alpha})}\{(1-|z|)_{2}F_{1}(2+\alpha,$$\alpha\alpha$

for $z\in U$ .
Letting;($\alpha$$2+\alpha+\dot{J}\alpha,$

;

|z|)

$|z|)+ \frac{4|z|}{2+\alpha}2F1(_{3+\alpha}\alpha,\alpha$$\}$; $|z|)\}$$\blacksquare$$\beta=\alpha$in Corollary 2.4, we have

Corollary 3.4. let $0\leq\alpha<1$ , $m=0,1$ and $\alpha+m<2$ . If $f(z)\in A(1,0)$ , then$|D_{z}^{m+\alpha}f(z)| \leq\frac{|z|^{1-\alpha-m}}{\Gamma(2-\alpha-m)(1-|z|)^{3+\alpha+m}}$$\mathrm{x}\{$$fo$ ’

(1

一

0く

$|z|$

$|z|$

) $2F1(-\alpha-m2-\alpha’$ $-\alpha-m-m$ ; $|z|)+ \frac{4|z|}{2-\alpha-m}2F1$ $(-\alpha-m3-\alpha’$ $-\alpha-m-m$ ; $|z|)\}$$<1$(3.13). If $f(z)\in A(0,0)$ , then$|D_{z}^{m+\alpha}f(z)| \leq\frac{|z|^{1-\alpha-m}}{\Gamma(2-\alpha-m)}2F1(2-\alpha-m2,2$; $|z|)$(3.13)

for $0<|z|<1$ .
Acknowledgements
The present work is partly supported by the Science Promotion Fund from the Japanby Bulgarian
Private School Promotion Foundation and Grant MM 1305/2003
Ministry of Education and Science of Bulgaria, NSF.
$(^{n}\mathrm{F}\mathrm{C}\mathrm{A}\mathrm{A}^{\mathrm{n}})$References[1] L. de Branges: Aproof of the Bieberbach conjecture, Acta Math. 154 (1985), 137-152.
[2] A. Erd\’elyi, W. Magnus, F. Oberhettinger and R.P. Soni: Higher Transcendental hnctions, Vol-s 1-3, McGraw-Hill, New York, 1953.
[3] S. Owa: On the distortion theorems, I, Kyungpook Math. J. 18 (1978), 53-59.

[4] C. Pommerenke: Univalent Functions, Vandenhoeck and Ruprecht, GStingen, 1975.
[5] M. Saigo: Aremark on integral operators involving the Gauss hypergeometric functions, Math. Rep. College General Ed. Kyushuhu Univ. 11 (1978), 135-143.
[6] H.M. Srivastava: Generalized hypergeometric functions with integral parameter differences, Nederi. Akad. Wetensch. Indag. Math. 35 (1973), 38-40.
[7] H.M. Srivastava, M. Saigo and S. Owa: Aclass of distortion theorems involving certain
operators of fractional calculus of starlike functions, J. Math. And. and Appl. 131(1988), 412-420