数理解析研究所講究録
1062 巻 1998 年 47-52

47

INTEGRAL MEANS OF THE FRACTIONAL DERIVATIVE OFUNIVALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS..

Yong Chan Kim and Jae Ho Choi

[Yeungnam University][崔宰豪, 福岡大学]

ABSTRACT. By using the definition of fractional derivative (cf., [2]), we investigate thesharp integral means inequalities of the fractional derivatives of univalent functions withnegative coefficients and extend the sharp results of H. Silverman [5, Theorem 2.2].

1. Introduction and Definitions
Let $A$ denote the class of $f(z)$ normalized by(1.1)$f(z)=z+ \sum_{k=2}^{\infty}a_{k}z^{k}$,

and $|z|<1$ }. Also, let $S$ denote
which are analytic in the open unit disk{
the class of all functions in $A$ which are univalent in . Then a function $f(z)$ belonging
if and only if
to the class $S$ is said to be in the class$\mathcal{U}=$$z:z\in \mathbb{C}$$\mathcal{U}$$\mathcal{K}$(1.2)

We denote by${\rm Re}(1+ \frac{zf^{\prime/}(z)}{f(z)},)>0$$\mathcal{T}$.

the subclass of $S$ whose functions may be represented by(1.3)$f(z)=z-k \sum_{=2}a_{k}z^{k}\infty$

Silverman [4] showed thatthat the extreme points of(1.4)$(z\in \mathcal{U})$$f_{1}(z)=z$$f$of the form (1.3) is in$\mathcal{T}$$(a_{k}\geq 0)$$\mathcal{T}$.

if and only ifareand$f_{m}(z)=z-z^{m}/m,$ $m=2,3,$.
1991 Mathematics Subject Classification.
Key words and phrases. fractional derivative, univalent, integral means.
This work was supported by KOSEF, BSRI-98-1401 and TGRC-KOSEF.
$30\mathrm{C}45$$\cdots$$\sum_{k=2}^{\infty}ka_{k}\leq 1$, and48,
if and only if
Further a function of the form (1.3) is inand that the extreme points of are $g_{1}(z)=z$ and $g_{2}(z)=z-Zm/m^{2}(m=2,3, \cdots)$ .
For analytic functions $g(z)$ and $h(z)$ with $g(\mathrm{O})=h(0),$ $g(z)$ is said to be subordinate
to $h(z)$ if there exists an analytic function $w(z)$ so that $w(\mathrm{O})=0,$ $|w(z)|<1(z\in \mathcal{U})$and $g(z)=h(w(z))$ , we denote this subordition by $g(z)\prec h(z)$ .
Many essentially equivalent definition of fractional calculus (that is, fractional derivatives and fractional integrals) have been given in the literature (cf., $e.g.,$ $[3],$and [7] . We find it to be convenient to recall here the following definition which wereused recently by Owa [2] (and by Srivastava and Owa [7]).
$\sum_{k=2}^{\infty}k^{2}ak\leq 1$$C=\mathcal{T}\cap \mathcal{K}$

$f$

$C$

$[6, \mathrm{p}45]$$)$

Definition 1. The fractional derivative of order(1.5)$\lambda$

is defined, for a function $f(z)$ , by

$D_{z}^{\lambda}f(z)= \frac{1}{\Gamma(1-\lambda)}\frac{d}{dz}\int_{0}^{z}\frac{f(z)}{(z-()^{\lambda}}d($$(0\leq\lambda<1)$,

where $f(z)$ is an analytic function in a simply-connected region of the -plane containing
is removed by requiring for $\log(z-()$ to be
the origin, and the multiplicity ofreal for $z>\zeta$ .
$z$$(z-\zeta)-\lambda$Definition 2. Under the hypotheses of Definition 1, the fractional derivative of orderis defined by$n+\lambda$$D_{z}n+ \lambda f(z)=\frac{d^{n}}{dz^{n}}D\lambda zf(z)$$(1.6\rangle$$(0\leq\lambda<1;n\in \mathbb{N}_{0}:=\{\mathrm{o}, 1,2, \cdots\})$.

In [5] it is proven that
(1.7)

for all$\int_{0}^{2\pi}|f(re^{i\theta})|^{\beta}d\theta\leq\int_{0}^{2\pi}|f2(re)i\theta|^{\beta}d\theta$$f\in \mathcal{T},$$\beta>0$

and $0<r<1$ . In this paper, by using the fractional derivative, weprove that
(1.8)$\int_{0}^{2\pi}|D_{z}^{\lambda}f(\Gamma e^{i}\theta)|^{\beta}d\theta\leq\int_{0}^{2\pi}|D_{z}^{\lambda}f_{2}(re)i\theta|^{\beta}d\theta$

for all$\beta>0,0<r<1$ and $0\leq\lambda<1$ . We also obtain the integral means.
inequality for $D_{z}^{n+\lambda}f(Z)(n=1,2)$ if$f\in \mathcal{T},$$f\in C(\mathrm{o}\mathrm{r}\mathcal{T})$2. Main Results

The following result will be required in our investigation.

49

Lemma. (Littlewood [1]) If$0<r<1$ ,
(2.1)

$f$

and

$g$

are analytic in$\mathcal{U}$with $g\prec f$ , then, for $\beta>0$ and.

$\int_{0}^{2\pi}|g(re)i\theta|^{\beta}d\theta\leq\int_{0}^{2\pi}|f(re^{i\theta})|^{\beta}d\theta$

Applying the above lemma, we prove
Theorem 1. Let $\beta>0$ and$0<r<1$ ,(i)(ii)$f_{2}(z)$is defined by (1.4). If$f\in \mathcal{T}$, then for$z=re^{i\theta}$

and$(0\leq\lambda<1)$$\int_{0}^{2\pi}|D^{\lambda}zf(_{Z)}|^{\beta}d\theta\leq\int_{0^{2\pi}}|D_{z}^{\lambda}f_{2}(Z)|^{\beta}d\theta$$\int_{0}^{2\beta\pi}\pi|D_{z}^{2}+\lambda f(z)|d\theta\leq\int_{0^{2\beta}}|D_{z}^{2\lambda}+f_{2}(z)|d\theta$$(0<\lambda<1)$ .

Proof. We prove (i). The proof of (ii) is similar and will be omitted. If $f(z)=z-$$\sum_{k=2}^{\infty k}a_{k^{Z}}(a_{k}\geq 0)$, then$D_{z}^{\lambda}f(z)= \frac{z^{1-\lambda}}{\Gamma(2-\lambda)}(1-\sum_{k=2}^{\infty}\Phi(k)ka_{k}z^{k1}-)$,

where(2.2)

Note that$\Phi(k^{\wedge})=\frac{\Gamma(k)\mathrm{r}(2-\lambda)}{\Gamma(k+1-\lambda)}$$\Phi(k)$$(k\geq 2)$.

is a non-increasing function of ,
$k$

(2.3)

$0< \Phi(k)\leq\Phi(2)=\frac{1}{2-\lambda}$

.

Since
$D_{z}^{\lambda}f_{2}(z)= \frac{z^{1-\lambda}}{\Gamma(2-\lambda)}(1-\frac{1}{2-\lambda}z)$,

we must show that$\int_{0}^{2\pi}|1-\sum_{k=2}^{\infty}\Phi(k)ka_{k}zk-1|^{\beta}d\theta\leq\int_{0}^{2\pi}|1-\frac{1}{2-\lambda}Z|\beta d\theta$

By Lemma, it sufficies to prove that
$1- \sum\Phi(\infty k)kakzk-1\prec 1-\frac{1}{2-\lambda}z$$k=2$.

.

50Setting
(2.4)

1–

$\sum_{k=2}^{\infty}\Phi(k)ka_{k}Zk-1=1-\frac{w(z)}{2-\lambda}$.

From (2.3) and (2.4), we obtain$|w(_{Z)|} \leq|_{k=2}\sum^{\infty}(2-\lambda)\Phi(2)k^{\wedge}akz^{k-}1|\leq|z|\sum_{=k2}^{\infty}kak\leq|z|$.

This completes the proof of (i).
Remark. If $\lambda=0$ in (i) of Theorem 1, then it would immediately yield the result ofSilverman [5, Theorem 2.2].
For the fractional derivative of order$1+\lambda$

Theorem 2. If $f\in C$ and $\beta>0$ , then for(i)(ii), we have$z=re^{i\theta}$$\int_{0}^{2\pi}|D_{z}^{1+\lambda}f(_{Z})|\beta d\theta\leq\int^{2\pi}0|D_{z}^{1+\lambda}f_{2(Z)}|^{\beta}d\theta$$\int_{0}^{2\pi}|D_{z}^{1+\lambda}f(Z)|^{\beta}d\theta\leq\int_{0}^{2\pi}|D_{z}^{2+\lambda}g2(z)|\beta d\theta$

and $0<r<1$ ,$(0\leq\lambda<1)$$(0\leq\lambda\leq 2/3)$.

Proof. (i) From the definition (1.6), we have(2.5)$D_{z}^{1+\lambda}f(Z)= \frac{z^{-\lambda}}{\Gamma(1-\lambda)}(1-\sum_{k=2}^{\infty}\Psi(k)k(k-1)ak^{Z})k-1$

where$(k\geq 2)$.

$D_{z}^{1+\lambda}f_{2}(_{Z})= \frac{z^{-\lambda}}{\Gamma(1-\lambda)}(1-\frac{1}{1-\lambda}z)$,$\Psi(k)=\frac{\Gamma(k^{\wedge}-1)\Gamma(1-\lambda)}{\Gamma(k-\lambda)}$Note that $0<\Psi(k)\leq\Psi(2)=1/(1-\lambda)$ .
Since

it suffices to show that
1– $\sum_{k=2}^{\infty}\Psi(k)k(k-1)akz^{k}-1\prec 1-\frac{1}{1-\lambda}z$ .

,51Setting,$1- \sum_{k=2}^{\infty}\Psi(k)k(k-1)a_{k}Zk-1=1-\frac{w(z)}{1-\lambda}$$|w(_{Z)|} \leq|\sum_{k=2}^{\infty}k(k-1)a_{k}Zk-1|\leq|z|\sum_{=k2}k^{2}a_{k}\infty\leq|z|$.

By Lemma, the proof of (i) is completed.
(ii) Making use of (1.6) and (2.5), we obtain$D_{z}^{1+\lambda}f(Z)= \frac{z^{-\lambda}}{\Gamma(1-\lambda)}(1-\sum_{k=2}^{\infty}\Theta(k)k2k-a_{k}z1\mathrm{I},$

where$\Theta(k-)=\frac{\Gamma(k)\Gamma(1-\lambda)}{k^{\wedge}\Gamma(k-\lambda)}$$(k\geq 2)$.

We note that $0<\Theta(k-)\leq\Theta(2)=1/2(1-\lambda)$ for $0\leq\lambda\leq 2/3$ . Thus the proof of (ii) is
much akin to that of (i), and we omit the details involved.
$0\leq\alpha<1$ , the subclasses ofthat are, respectively,andDenote bythatshowed[4],
Silverman
starlike of order and convex of order . In
if and only if $\sum_{k=2}^{\infty}((k--\alpha)/(1-\alpha))ak\leq 1$ and $f\in C(\alpha)$ if and only if
are $h_{m}(z)=$$\alpha)/(1-\alpha))a_{k}\leq 1$ .
andIn addition, the extreme points of$z-((1-\alpha)/(m-\alpha))z^{m}$ and $k_{m}(z)=z-((1-\alpha)/m(m-\alpha))_{Z^{m}}$ for $m\geq 2$ .
, the proof is much akin to that of Theorem 1 andandFor the cases ofTheorem 2, and we omit the details involved.
$\mathcal{T}$$C(\alpha),$$\mathcal{T}^{*}(\alpha)$$f\in \mathcal{T}^{*}(\alpha)$$\alpha$$\alpha$$\sum_{k=2}^{\infty}(k(k-$$T^{*}(\alpha)$$\mathcal{T}^{*}(\alpha)$

Theorem 3. (i) If$C(\alpha)$$C(\alpha)$$f\in \mathcal{T}^{*}(\alpha)$

and $\beta>0$ , then for $0<r<1$ ,$\int_{0}^{2\pi}|D_{z}^{\lambda}f(re)i\theta|^{\beta}d\theta\leq\int_{0}^{2\pi}|D_{z}^{\lambda}h_{2(e^{i\theta})|^{\beta}d\theta}r$$(0\leq\lambda<1)$and$\int_{0}^{2\pi}|D_{z}^{2+\lambda}f(re^{i})\theta|^{\beta}d\theta\leq\int_{0}^{2\pi}|D_{z}^{2+\lambda}h_{2}(re^{i\theta})|^{\beta}d\theta$$(0<\lambda<1)$(ii) If $f\in C(\alpha)$ and $\beta>0$ , then for $0<r<1_{f}$$\int_{0}^{2\pi}|D_{z}^{\lambda}f(re^{i}\theta)|^{\beta}d\theta\leq\int_{0}^{2\pi}|D_{z}^{\lambda}k_{2}(re^{i})\theta|^{\beta}d\theta$$(0\leq\lambda<1)$

,.

52

$(0\leq\lambda<1)$$\int_{0}^{2\pi}|D_{z}1+\lambda f(rei\theta)|^{\beta}d\theta\leq\int_{0}^{2\pi}|D_{z}^{1+\lambda}h2(re^{i})\theta|^{\beta}d\theta$$\int_{0}^{2\pi}|D^{1\lambda}f(re^{i\theta})\mathcal{Z}^{+}|\beta d\theta\leq\int_{0}^{2\pi}|D^{1}+\lambda k_{2}(re^{i})z|\theta\beta d\theta$,$(0\leq\lambda\leq 2/3)$

and$\int_{0}^{2\pi}|D_{z}^{2}+\lambda f(rei\theta)|\beta d\theta\leq\int_{0}^{2\pi}|D^{2\lambda}+k2(re^{i\theta})z|\beta d\theta$$(0<\lambda<1)$ .

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4. H. Silverman, Univalent functions wnth negative coefficients, Proc. Amer. Math. Soc. 51 (1975),
109-116.
5. H. Silverman, Integral mean8 for univalent functions vnth negative coefficients, Houston J. Math.
23 (1997), 169-174.
6. H.M. Srivastava and R.G. Buschman, Theory and Applications of Convolution Integral Equations,, 1992.
Kluwer Academic,
7. H.M. Srivastava and S. Owa, An application of the fractional derivative, Math. Japon. 29 (1984),
383-389.
$\mathrm{D}\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{t}-\mathrm{B}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{n}-\mathrm{L}\mathrm{o}\mathrm{n}\mathrm{d}_{0}\mathrm{n}$Yong Chan Kim
Department of Mathematics
Yeungnam University
Gyongsan 712-749
Korea

Jae Ho Choi
Department of Applied Mathematics
Fukuoka University

Fukuoka 814-0180
Japan