数理解析研究所講究録
1112 巻 1999 年 36-43

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ON SOME ANGULAR ESTIMATES-CONVEX FUNCTIONS
OF$\mathrm{C}\mathrm{L}\mathrm{O}\mathrm{S}\mathrm{E}-\mathrm{T}\mathrm{O}$AKIRA IKEDA AND MEGUMI SAIGOABSTRACT. The paper is devoted to generalizing the results by Libera [4],Pommerenke [6] and Ponnusamy and Karunakaran [7] relating to properties of close-toconvex functions.

$\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{G}\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{o}\mathrm{r}[5])$1. IntroductionLet $p\in N=\{1,2,3, \cdots\}$ and $A(p)$ denote the class of functions$f(z)=z^{p}+ \sum_{k=p+1}^{\infty}a_{k}z^{k}$which are analytic in the unit disk $\mathcal{U}=\{z:|z|<1\}$ . A function $f(z)\in A(p)$ is called-valently starlike if$p$${\rm Re} \{\frac{zf(z)}{f(z)},\}>0$

in$\mathcal{U}$.

We denote by $S^{*}(p)$ the subclass of $A(p)$ consisting of -valently starlike functions.
Further, a function in $A(p)$ is said to be p–valently convex if$p$in$1+{\rm Re} \{\frac{zf’’(z)}{f’(z)}\}>0$$\mathcal{U}$.

Let $C(p)$ denote the subclass of $A(p)$ of such -valently convex functions in . A function$f(z)\in A(p)$ is said to be p–valently close-to-convex if there is a function $g(z)\in C(p)$such that$\mathcal{U}$$p$${\rm Re} \{\frac{f’(z)}{g’(z)}\}>0$

in$\mathcal{U}$.

the class of -valently close-to-convex functions. As is well
We shall denote byknow, we have the inclusions$\mathcal{K}(p)$$p$$C(p)\subset S^{*}(p)\subset \mathcal{K}(p)$.

Now, we define the subordination. Let $f(z)$ and $g(z)$ be analytic in , with. Suppose $f(z)$ is univalent, and the range of by $g(z)$ is contained in that of $f(z)$ .
Then we say the function $g(z)$ subordinates to $f(z)$ and write $g(z)\prec f(z)$ .
$\mathcal{U}$$\mathcal{U}$$g(\mathrm{O})$1991 Mathematics Subject Classification.

$f(\mathrm{O})=$$30\mathrm{C}45$.
Typeset by$A_{\mathcal{M}}S- \mathrm{I}\mathrm{E}\mathrm{X}$37

A. IKEDA AND M. SAIGO

Theorem A. [3] Let $f(z)\in A(p)$ . Let $g(z)\in S^{*}(p)$ satisfy${\rm Re} \{,\frac{f(z)}{g(z)},\}>0$

in${\rm Re} \{\frac{f(z)}{g(z)}\}>0$

in,$\mathcal{U}$then we have$\mathcal{U}$.

Theorem A was proved by Sakaguchi [3], which is generalized by Libera [4],gor [5], Pommerenke [6], and Ponnusamy and Karunakaran [7].

The generalization ofof Libera [4]:$\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{G}\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{o}\mathrm{r}[5]$$\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{G}\mathrm{r}e_{\lrcorner}-$is the following, which is quite similar to thatTheorem B. [5, Lemma 2] Suppose that functions $f(z)$ and $g(z)$ are analytic in with$f(\mathrm{O})=g(\mathrm{O})=0$ , and $g(z)$ maps
onto a region which is starlike with respect to theorigin. Let $0\leq\gamma<1$ . If$\mathcal{U}$$\mathcal{U}$${\rm Re} \{,\frac{f’(z)}{g(z)}\}>\gamma$in${\rm Re} \{\frac{f(z)}{g(z)}\}>\gamma$in,$\mathcal{U}$then

Likewise, if$\mathcal{U}$.

.

${\rm Re} \{\frac{f’(z)}{g’(z)}\}<\gamma$in${\rm Re} \{\frac{f(z)}{g(z)}\}<\gamma$in,$\mathcal{U}$then$\mathcal{U}$.

In [6], Pommerenke obtained the following theorem.
Theorem C. [6, Lemma 1] Let $f(z),$ $g(z)\in A(p)$ . For $0\leq\alpha\leq 1$ ,$| \arg\{\frac{f’(z)}{g’(z)}\}|\leq\frac{\pi}{2}\alpha$in$\mathcal{U}$,then$| \arg\{\frac{f(z_{2})-f(z_{1})}{g(z_{2})-g(z_{1})}\}|\leq\frac{\pi}{2}\alpha$for$z_{1},$$z_{2}\in \mathcal{U}$.

In [7], Ponnusamy and Karunakaran lead the next theorem.

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ON SOME ANGULAR ESTIMATES OF CLOSE-TO-CONVEX FUNCTIONS

Theorem D. [7, Corollary 2] Let $p\geq 1,$ $k\geq 1,$ $\beta<1$ and $0\leq\delta<1/p$ . If $f(z),g(z)\in$$A(p)$ and $g(z)$ satisfies${\rm Re} \{\frac{g(z)}{zg(z)},\}>\delta$,then${\rm Re} \{,\frac{f’(z)}{g(z)}\}>\beta$

implies${\rm Re} \{\frac{f(z)}{g(z)}\}>\frac{2\beta+k\delta}{2+k\delta}$

Theorem$\mathrm{D}$.

may be regarded as a generalization of the results of Theorems A and B.

In 1995, Nunokawa obtained the next two theorems.
Theorem E. [8, Theorem 1] Let $f(z)\in A(p),$ $g(z)\in S^{*}(p),$ $0<\alpha\leq 1$ andnumber. Suppose that$| \arg\{,\frac{f’(z)}{g(z)}-\beta\}|<\frac{\pi}{2}\alpha$in$| \arg\{\frac{f(z)}{g(z)}-\beta\}|<\frac{\pi}{2}\alpha$

in$\beta$

be a real,$\mathcal{U}$then we have$\mathcal{U}$.

Theorem F. [8, Theorem 2] Let $f(z)\in A(p),$ $g(z)\in S^{*}(p)$ , where $0<\alpha\leq 1$ and$\beta>1$ . Suppose that$| \arg\{\beta-,\frac{f’(z)}{g(z)}\}|<\frac{\pi}{2}\alpha$in$\mathcal{U}$$| \arg\{\beta-\frac{f(z)}{g(z)}\}|<\frac{\pi}{2}\alpha$in$\mathcal{U}$,then we have

$or$

in

$\pi-\frac{\pi}{2}\alpha<\arg\{\frac{f(z)}{g(z)}-\beta\}<\pi+\frac{\pi}{2}\alpha$$\mathcal{U}$.

Remark 1. Theorem is a generalization of Theorem , the first half of Theorem
and Theorem , while Theorem is a generalization of the second half of Theorem B.
$\mathrm{A}$$\mathrm{E}$$\mathrm{C}$$\mathrm{F}$$\mathrm{B}$39

A. IKEDA AND M. SAIGO

2. Preliminaries
In this paper, we need the following lemmas.

Lemma 1. [10] Let $p(z)$ be analytic inand suppose that there exists a point$\mathcal{U}$withsuch thatand $p(z)\neq 0$ in$p(\mathrm{O})=1$$\mathcal{U}$. Let $\beta>0$$z_{0}\in \mathcal{U}$for$| \arg\{p(z)\}|<\frac{\pi}{2}\beta$$|z|<|z_{0}|$and$| \arg\{p(z_{0})\}|=\frac{\pi}{2}\beta$.

Then we have$\frac{z_{0}p(z_{0})}{p(z_{0})},=ik\beta$,

where$k\geq 1$

when$\arg\{p(z_{0})\}=\frac{\pi}{2}\beta$$k\leq-1$

when$\arg\{p(z_{0})\}=-\frac{\pi}{2}\beta$,

and$p(z_{0})^{1/\beta}=\pm ia$

Lemma 2. Let
and $p(z)\neq 0$ in(1)$\alpha$$\mathcal{U}$,$a>0$ .

be a positive real number and let $p(z)$ be analytic in. Let-land suppose thatwith$\mathcal{U}$$\leq\delta<\lambda\leq 1$

in$| \arg\{p(z)+,\frac{g(z)}{g(z)}p’(z)\}|<\frac{\pi}{2}\alpha$$\mathcal{U}$$or$in$p(z)+, \frac{g(z)}{g(z)}p’(z)\prec(\frac{1+z}{1-z})^{\alpha}$$\mathcal{U}$,where $g(z)$ belongs to $S^{*}(p)$ and satisfies(2)$\frac{g(z)}{zg’(z)}\prec\frac{1}{p}\frac{1+\lambda z}{1+\delta z}$.

Then for $\beta>0$ being determined by(3)$\alpha=\beta+\frac{2}{\pi}\tan^{-1}\{\frac{(1-\lambda)\{(\lambda-\delta)\beta+p(1-\lambda)(1-\delta^{2})\}}{p(1-\delta)(\lambda-\delta)}\}$

we have$| \arg\{p(z)\}|<\frac{\pi}{2}\beta$

in$\mathcal{U}$.

,$p(\mathrm{O})=1$

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ON SOME ANGULAR ESTIMATES OF CLOSE-TO-CONVEX FUNCTIONSProof. Suppose that there exists a point$z_{0}\in \mathcal{U}$such thatfor$| \arg\{p(z)\}|<\frac{\pi}{2}\beta$$|z|<|z_{0}|$and$| \arg\{p(z_{0})\}|=\frac{\pi}{2}\beta$.

Then, from Lemma 1, we have$\frac{z_{0}p’(z_{0})}{p(z_{0})}=ik\beta$,

where,$k\geq 1$

when$\arg\{p(z_{0})\}=\frac{\pi}{2}\beta$$k\leq-1$

when$\arg\{p(z_{0})\}=-\frac{\pi}{2}\beta$

and$p(z_{0})^{1/\beta}=\pm ia$,$a>0$ .

Then it follows that$\arg\{p(z_{0})+,\frac{g(z_{0})}{g(z_{0})}p’(z_{0})\}=\arg\{p(z_{0})\}[1+\frac{z_{0}p’(z_{0})}{p(z_{0})}\frac{g(z_{0})}{z_{0}g(z_{0})},]$$= \arg\{p(z_{0})\}[1+ik\beta\frac{g(z_{0})}{z_{0}g^{l}(z_{0})}]$$=\arg\{p(z_{0})\}(A+iB)$ .

Here real constants$A$

and$B$

can be estimated by virtue of the assumption (2) such as$A \leq 1+\frac{1}{p}\frac{\lambda-\delta}{1-\delta^{2}}k\beta$(4)$B \geq\frac{1}{p}\frac{1-\lambda}{1-\delta}k\beta$

Note that the right hand side of (4) is positive.

.

,

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A. IKEDA AND M. SAIGO

When $\arg\{p(z_{0})\}=\pi\beta/2$ , we have$\arg\{p(z_{0})+,\frac{g(z_{0})}{g(z_{0})}p’(z_{0})\}=\arg\{p(z_{0})\}(A+iB)$$\geq\frac{\pi}{2}\beta+\tan^{-1}\{\frac{\frac{1}{p}\frac{1-\lambda}{1-\delta}k\beta}{1+\frac{1}{p}\frac{\lambda-\delta}{1-\delta^{2}}k\beta}\}$$= \frac{\pi}{2}\beta+\tan^{-1}\{\frac{(1-\lambda)\{(\lambda-\delta)k\beta+p(1-\lambda)(1-\delta^{2})\}}{p(1-\delta)(\lambda-\delta)}\}$$\geq\frac{\pi}{2}\beta+\tan^{-1}\{\frac{(1-\lambda)\{(\lambda-\delta)\beta+p(1-\lambda)(1-\delta^{2})\}}{p(1-\delta)(\lambda-\delta)}\}$$= \frac{\pi}{2}[\beta+\frac{2}{\pi}\tan^{-1}\{\frac{(1-\lambda)\{(\lambda-\delta)\beta+p(1-\lambda)(1-\delta^{2})\}}{p(1-\delta)(\lambda-\delta)}\}]$$\pi$$=\overline{2}\alpha$.

On the other hand, when $\arg\{p(z_{0})\}=-\pi\beta/2$ , we have$\arg\{p(z_{0})+,\frac{g(z_{0})}{g(z_{0})}p’(z_{0})\}=\arg\{p(z_{0})\}(A+iB)$$\leq-\frac{\pi}{2}[\beta+\frac{2}{\pi}\tan^{-1}\{\frac{(1-\lambda)\{(\lambda-\delta)\beta+p(1-\lambda)(1-\delta^{2})\}}{p(1-\delta)(\lambda-\delta)}\}]$$\pi$$=-\alpha\overline{2}$.

These contradict (1), which completes the proof of Lemma 2.

Remark 2. Note that when$\lambda=1,$

Remark 3. The existence ofeasily.

$\beta$$\beta=\alpha$from the equation (1).

satisfying (3) for any positive$\alpha$can be certificated

3. Main results
Theorem 1. Let
and$\gamma$

be a real number and $0<\alpha\leq 1$ . Let $f(z)\in A(p),$ $g(z)\in S^{*}(p)$$\frac{g(z)}{zg(z)},\prec\frac{1}{p}\frac{1+\lambda z}{1+\delta z}$for-l(5)$\leq\delta<\lambda\leq 1$

and suppose that$| \arg\{,’\frac{f(z)}{g(z)}-\gamma\}|<\frac{\pi}{2}\alpha$in$\mathcal{U}$.

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ON SOME ANGULAR ESTIMATES OF CLOSE-TO-CONVEX FUNCTIONSThen for $\beta>0$ being determined by (3)$\mathrm{r}ve$

havein$| \arg\{\frac{f(z)}{g(z)}-\gamma\}|<\frac{\pi}{2}\beta$$\mathcal{U}$.

Proof. Let us put$p(z)= \frac{1}{1-\gamma}\{\frac{f(z)}{g(z)}-\gamma\}$.

Then we have$p(z)+, \frac{g(z)}{g(z)}p’(z)=\frac{1}{1-\gamma}\{,\frac{f(z)}{g(z)},-\gamma\}$.

Applying Lemma 2 for this $p(z)$ , we obtain the required result.
Remark 4. Theorem 1 is a revision of Theorem$\mathrm{E}$in view of Remark 2.

Theorem 2. Let $\gamma>1$ and $0<\alpha\leq 1$ . Let $f(z)\in A(p),$ $g(z)\in S^{*}(p)$ . For-l$\lambda\leq 1$$\leq\delta<$

we assume$\frac{g(z)}{zg(z)},\prec\frac{1}{p}\frac{1+\lambda z}{1+\delta z}$

and suppose that$| \arg\{\gamma-\frac{f’(z)}{g’(z)}\}|<\frac{\pi}{2}\alpha$in$\mathcal{U}$in$\mathcal{U}$.

Then for $\beta>0$ being determined by (3) we have$| \arg\{\gamma-\frac{f(z)}{g(z)}\}|<\frac{\pi}{2}\beta$

$or$

in

$\pi-\frac{\pi}{2}\beta<\arg\{\frac{f(z)}{g(z)}-\gamma\}<\pi+\frac{\pi}{2}\beta$$\mathcal{U}$.

Proof. Let us put$p(z)= \frac{1}{\gamma-1}\{\gamma-\frac{f(z)}{g(z)}\}$.

Then we have$p(z)+, \frac{g(z)}{g(z)}p’(z)=\frac{1}{\gamma-1}\{\gamma-\frac{f’(z)}{g’(z)}\}$,

which yields the result of the present theorem.
Remark 5. Theorem 2 is better than Theorem$\mathrm{F}$, as we noted in Remark 3.

Remark 6. In case of $\lambda=1,$ $\alpha=\beta=1$ and $\gamma=0$ , Theorem 1 is equivalent toTheorem A.

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A. IKEDA AND M. SAIGOREFERENCES
1. W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J. 1 (1952), 169-185.
2. P. L. Duren, Univalent Functions, Springer-Verlag, New York, Berlin, Heidelberg (1983).
3. K. Sakaguchi, On a certain univalent mapping) J. Math. Soc. Japan 11 (1959), 72-75.
4. R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1965),755-758.
5. T. H. MacGregor, A subordination for convex functions of order , J. London Math. Soc. (2) 9$\alpha$(1975), 530-536.
6. Ch. Pommerenke, On close-to-convex analytic functions, Trans. Amer. Math. Soc. 114 (1965),

176-186.
7. S. Ponnusamy and V. Karunakaran, Differential subordination and conformal mappings, Complex
Variables 11 (1989), 79-86.
8. M. Nunokawa, On some angular estimates of analytic functions, Math. Japon. 41 (1995), 447-452.
9. M. Nunokawa, On properties of non-Carathe’odory functions, Proc. Japan Acad. 68 (1992), 152153.
10. M. Nunokawa, On the order of strongly starlikeness of strongly convex functions, Proc. JapanAcad. 69 (1993), 234-237.
AKIRA IKEDA:
MEGUMI SAIGO:DEPARTMENT OP APPLIED MATHEMATICS, FUKUOKA UNIVERSITY,
8-19-1 NANAKUMA, JONAN-KU, FUKUOKA, 814-0180, JAPAN-mail address:sm. fukuoka-u. ac. jp,$E$$\mathrm{a}i\mathrm{k}\mathrm{e}\mathrm{d}\mathrm{a}\emptyset \mathrm{s}\mathrm{f}.$$\mathrm{m}\mathrm{s}\mathrm{a}\mathrm{i}\mathrm{g}\mathrm{o}\Phi \mathrm{f}\mathrm{u}\mathrm{k}\mathrm{u}\mathrm{o}\mathrm{k}\mathrm{a}-\mathrm{u}.$ac. jp