数理解析研究所講究録 1341 巻 2003 年 1-11

1

ON THE FEKETE-SZEGO PROBLEM FOR STRONGLY-LOGARITHMIC QUASICONVEX FUNCTIONS$\alpha$NAK EUN CHO AND SHIGEYOSHI OWA
andABSTRAOT. The purpose of the present paper is to introduce the classes, respectively, of normalized strongly a-logarithmic convex and quasiconvex func$\mathcal{M}^{\alpha}(\beta)$$Q^{\alpha}(\beta)$

tions of order in the open unit disk and to obtain sharp Ebket&Szeg\={o} inequalities for.
andfunctions belonging to the classes be the class$\beta$$Q^{\alpha}(\beta)$$\mathcal{M}^{a}(\beta)$

1. Introduction
Let $S$ denote the class of analytic functions$f(z)$ $=z$$f$of the form

(1.1)$+ \sum_{n=2}^{\infty}a_{n}z^{n}$which are univalent in the open unit disk $\mathcal{U}=\{z:|z| <1\}$ . Aclassical theorem ofFekete and Szeg\"o [8] states that for $f\in S$ given by (1.1),$|a_{3}-\mu a_{2}^{2}|\leq\{\begin{array}{l}3-4\mu \mathrm{i}\mathrm{f}\mu\leq 0\mathrm{l}+2e^{-2\mu/(1-\mu)}\mathrm{i}\mathrm{f}0\leq\mu\leq \mathrm{l}4\mu-3\mathrm{i}\mathrm{f}\mu\geq \mathrm{l}\end{array}$This inequality is sharp in the sense that for each there exists afunction in $S$
such that equality holds. Recently, Pfluger $[17,18]$ has considered the problem whenis complex. In the case of ,and , the subclasses of convex, starlike andclose-t0-convex functions, respectively, the above inequality can be improved $[10,11]$ .
Also, Darus and Thomas [5] studied the classof a-logarithmic convex functions. Furthermore,
and they also have solved the Fekete-Szeg\"o problem for the class
London [14] have extended the results of Abdel-Gawad and Thomas [1], Keogh and$\mu$$\mathrm{C}$$\mu$$S^{*}$$\mathcal{K}$$\mathcal{M}^{a}$$\mathcal{M}^{\alpha}$1991 Mathematics Subject Classification:.
Key words and phrases. Fekete-Szego problem, strongly
strongly a-logarithmic quasiconvex.
$30\mathrm{C}45$$a$-logarithmic convex, quasiconvex,

Typeset by$\mathcal{M}r\theta \mathrm{I}\mathrm{f}\mathrm{f}1$

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N. E. CHO AND S. OWA

Merkes [10] and Koepf $[11,12]$ to the classof strongly close-t0-convex functionsof order . Now we introduce new classes which incorporate well-known classes of
univalent functions.
$\mathcal{K}(\beta)$$\beta$Definition 1.1.
Afunction $f\in S$ given by (1.1) is said to be strongly logarithmic -convex of order if$\beta$$\alpha$$| \arg\{(\frac{zf’(z)}{f(z)})^{1-a}(\frac{(zf’(z))’}{f(z)},)^{\alpha}\}|\leq\frac{\pi}{2}\beta$(a $\geq 0;0<\beta\leq 1;z\in \mathcal{U}$).

(1.2)

Denote by
the class of strongly -logarithmic convex functions of order . The class
was introduced by Chiang [4]. In particular, the classes
andhave been extensively studied by Lewandowski, Miller
and Zlotkiewiez [13] and Bramnan and Kirwan [2](also, see [7,20]), respectively.
$\mathcal{M}^{\alpha}(\beta)$$\beta$$\alpha$$\mathcal{M}^{\alpha}(\beta)$$\mathcal{M}^{a}(1)=\mathcal{M}^{\alpha}$$\mathcal{M}^{0}(\beta)$

Definition 1.2. Afunction $f\in S$ given by (1.1) is said to be a-logarithmic
quasiconvex of order if there exists afunction
such that$g\in \mathrm{C}$$\beta$$| \arg\{(,\frac{f’(z)}{g(z)})^{1-\alpha}(\frac{(zf’(z))’}{g(z)},)^{\alpha}\}|\leq\frac{\pi}{2}\beta$$(\alpha,\beta\geq 0;z \in \mathcal{U})$

.

(1.3)

We denote by
the class of strongly -logarithmic quasiconvex functions oforder . Clearly,
andare the classes of close-t0-convex functions andquasiconvex functions introduced by Kaplan [9] and Noor [15](also, see [16]), respectively. Also we note that.
In the present paper, we derive sharp Fekete-Szeg\"o inequalities for functions belonging to the classes, which imply the results obtained by Abdeland
Gawad and Thomas [1], Darus and Thomas [5], Keogh and Merkes [10], Koepf,
and London [14].
$Q^{\alpha}(\beta)$$\beta$$\alpha$$Q^{0}(1)$$Q^{1}(1)$$Q^{0}(\beta)=\mathcal{K}(\beta)$$\Lambda 4^{\alpha}(\beta)$$Q^{\alpha}(\beta)$$[1\mathrm{I},12]$

2. Results

To prove our main results, we need the following

Lemma 2.1, Let

be analytic in$p(z)=1+p_{1}z+p_{2}z^{2}+\cdots$ . Then$p$$\mathcal{U}$and satisfy ${\rm Re}\{p(z)\}>0$ far$z\in \mathcal{U}$, with(2.1)$|p_{n}|\leq 2(n\geq 1)$and$|p_{2}- \frac{p_{1}^{2}}{2}|\leq 2-\frac{|p_{1}|^{2}}{2}$

.

(2.2)

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THE FEKETE-SZEG\"O PROBLEM

The inequality (2.1) was first proved by Carath\’eodory
and the inequality (2.2) can be found in [19, p.166].

$[3]$

(

$\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{o}$, see Duren [6, p. 41])

With the help of Lemma 2.1, we now deriveand be given by (11). Then for complex number
Theorem 2,1. Let$f\in \mathcal{M}^{\alpha}(\beta)$

$\mu$

,

$|a_{3}- \mu a_{2}^{2}|\leq\frac{\beta}{1+2\alpha}\max\{1$For each , there is a function in
Proof. Prom (1.2), we can write,$\mathcal{M}^{\alpha}(\beta)$$\mu$$\frac{|3(1+3\alpha)-4\mu(1+2\alpha)|\beta}{(1+\alpha)^{2}}\}$

such that equality holds.

$( \frac{zf’(z)}{f(z)})^{1-\alpha}(\frac{(zf’(z))’}{f(z)},)^{\alpha}=p^{\beta}(z)$

where

$p$

.

,

is given by Lemma 2.1. Equating coefficents, we obtain(2.3)$a_{2}= \frac{\beta}{1+\alpha}p_{1}$and.

$a_{3}= \frac{1}{4(1+2\alpha)}(\beta(\beta-1)p_{1}^{2}+2\beta p_{2}-(\alpha^{2}-7\alpha-2)(\frac{\sqrt p_{1}}{1+\alpha})^{2})$Then we have$a_{3}- \mu a_{2}^{2}=\frac{\beta}{2(1+2\alpha)}(p_{2}-\frac{p_{1}^{2}}{2})+\frac{(3+9\alpha-4\mu(1+2\alpha))\beta^{2}p_{1}^{2}}{4(1+2\alpha)(1+\alpha)^{2}}$

.

(2.4)

Hence (2.4) and Lemma 2.1 give$|a_{3}- \mu a_{2}^{2}|\leq\frac{\beta}{2(1+2\alpha)}(2-\frac{|p_{1}|^{2}}{2})+\frac{|3+9\alpha-4\mu(1+2\alpha)|\beta^{2}|p_{1}|^{2}}{4(1+2\alpha)(1+\alpha)^{2}}$.

$\leq\frac{\beta}{1+2\alpha}+\frac{\{|3+9\alpha-4\mu(1+2\alpha)|\beta^{2}-(1+\alpha)^{2}\beta\}|p_{1}|^{2}}{4(1+2\alpha)(1+\alpha)^{2}}$Therefore, by using$|p_{1}|\leq 2$, we haveif$k(\alpha)\leq*^{2}1+\alpha$,if$k( \alpha)\geq\frac{(1+\alpha)^{2}}{-\beta}$,$|a_{3}-\mu a_{2}^{2}|\leq\{$

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N. E. CHO AND S. OWA

where$k(\alpha)=|3(1+3\alpha)-4\mu(1+2\alpha)|$Equality is attained for functions in$\mathcal{M}^{\alpha}(\beta)$.

, respectively, given by$( \frac{zf’(z)}{f(z)})^{1-\alpha}(\frac{(zf’(z))’}{f(z)},)^{\alpha}=(\frac{1+z^{2}}{1-z^{2}})^{\beta}$(2.5)$( \frac{zf’(z)}{f(z)})^{1-\alpha}(\frac{(zf’(z))’}{f(z)},)^{\alpha}=(\frac{1+z}{1-z})^{\beta}$

(2.6)

and

Remark 2.1. It follows at once from (2.3) that $|a_{2}|\leq 2\beta/(1+\alpha)$ and Theorem
2.1 gives
if $(1+\alpha)^{2}\geq 3(1+3\alpha)\beta$,$|a_{3}|\leq\{$

if $(1+\alpha)^{2}\leq 3(1+3\alpha)\beta$,

The inequality forare sharp when$|a_{3}|$is sharp when is defined by (2.6) and the inequalities foris defined by (2.5) and (2.6), respectively.
$f$$|a_{2}|$$f$

Next, we consider the real number

Theorem 2.2. Let$f\in \mathcal{M}^{\alpha}(\beta)$$\mu$

as follows.

and be given by (1.1). Then for real number ,$\mu$$|a_{3}-\mu a_{2}^{2}|\leq\{\begin{array}{l}\frac{(3(1+3\alpha)-4(1+2\alpha)\mu)\beta^{2}}{\neg^{-}1\overline{+2}\alpha 7[1^{-}+\alpha)^{\mathcal{T}}},if\mu\leq\frac{3(1+3\alpha)\beta-(1+a)^{2}}{41\overline{1+2}\alpha\urcorner F^{-}}\mp 1\overline{2\alpha}\epsilon,if\frac{3(1+3a)\beta-(1+\alpha\}^{2}}{\neg 41+2\alpha\neg\overline{\beta}-}\leq\mu\leq\frac{3(1+S\alpha)\beta+(1+\alpha)}{4\zeta 1\overline{+2}a)\beta^{-}}’\frac{(4(1+2\alpha)\mu-3(1+\S\alpha))\beta^{2}}{(1+2\alpha)(1+\alpha)^{2}},\dot{l}f\mu\geq\frac{3(1+\S a)\beta+(1+a)^{l}}{4\Gamma 1+2a)\overline{\beta}}\end{array}$For each , there is a function insuch that equality holds in all cases.
Proof We consider two cases. At first, we suppose that $\mu\leq 3(1+3\alpha)/(4(1+2\alpha))$ .
Then (2.3) and Lemma 2.1 give$\mathrm{C}^{\alpha}(\beta)$$\mu$$|a_{3}- \mu a_{2}^{2}|\leq\frac{\beta}{2(1+2\alpha)}(2-\frac{|p_{1}|^{2}}{2})+\frac{(3+9\alpha-4\mu(1+2\alpha))\beta^{2}|p_{1}[^{2}}{4(1+2\alpha)(1+\alpha)^{2}}$.

$\leq\frac{\beta}{1+2\alpha}+\frac{((3+9\alpha-4\mu(1+2\alpha))\beta^{2}-(1+\alpha)^{2}\beta)|p_{1}|^{2}}{4(1+2\alpha)(1+\alpha)^{2}}$So, by using the fact that$|p_{1}|\leq 2$, we obtai

5
THE FEKETE-SZEG\"O PROBLEM

if$\mu\leq$$|a_{3}-\mu a_{2}^{2}|\leq\{$$\overline{1}+\overline{2\alpha}l$,if$\leq\mu\leq\frac{3(1+3\alpha)}{\frac{41+\mathit{2}\alpha}{}}$,

Equality is attained by choosing $p_{1}=p_{2}=2$ and $p_{1}=0$ , $p_{2}=2$ , respectively, in(2.3).
Next, we suppose that $\geq 3(1+3\alpha)/(4(1+2\alpha))$ . In this case, it follows again
from (2.3) and Lemma 2.1 that$\mu$$|a_{3}- \mu a_{2}^{2}|\leq\frac{\beta}{2(1+2\alpha)}(2-\frac{|p_{1}|^{2}}{2})+\frac{(4\mu(1+2\alpha)-(3+9\alpha))\beta^{2}|p_{1}|^{2}}{4(1+2\alpha)(1+\alpha)^{2}}$$\leq\frac{\beta}{1+2\alpha}+\frac{((4\mu(1+2\alpha)-(3+9\alpha))\beta^{2}-\beta(1+\alpha)^{2})|p_{1}|^{2}}{4(1+2\alpha)(1+\alpha)^{2}}$,and so, as in the first case, we have’$|a_{3}-\mu a_{2}^{2}|\leq\{\overline{1}+\overline{2a}\mathit{1}$

The results are sharp by choosing $p_{1}=0$ , $p_{2}=2$ and $p_{1}=2i$ , $p_{2}=-2$ , respectively,in (2.3).
If we take $\beta=1$ in Theorem 2.1 and Theorem 2.2, then we
Remark 2.2.
obtain the results by Darus and Thomas [5].
Finally, we prove

Theorem 2.3. Let
we have$f\in Q^{\alpha}(\beta)$

and be given by (1.1). Then for$\alpha\geq 0$and$3(2\alpha+1)|a_{8}-\mu a_{2}^{2}|\leq\{$

For each , there is a function in$\mu$$Q^{\alpha}(\beta)$

such that equality holds in all cases$\beta$$\geq 0$,

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N. E. CHO AND S. OWA

Proof.

Let$f\in Q^{\alpha}(\beta)$. Then it follows from (1.2) that we may write$(, \frac{f’(z)}{g(z)})^{1-\alpha}(\frac{(zf’(z))’}{g(z)},)^{\alpha}=p^{\beta}(z)$

,

(2.7)

where is convex and has positive real part. Let $g(z)=z+\mathrm{h}z^{2}+b_{3}z^{3}+\cdots$ andlet be given in the Lemma above. Then by comparing the coefficients of both sidesof (2.7), we obtain
$g$

$p$

$p$

$2(\alpha+1)a_{2}=\beta p_{1}+2b_{2}$and$3(2 \alpha+1)a_{3}=3b_{3}+\frac{2\alpha(1-\alpha)}{(\alpha+1)^{2}}b_{2}^{2}+\beta(p_{2}-\frac{1}{2}p_{1}^{2})$.

$+ \frac{\beta^{2}(3\alpha+1)}{2(\alpha+1)^{2}}p_{1}^{2}+\frac{2\beta(3\alpha+1)}{(\alpha+1)^{2}}p_{1h}$So, with$x= \frac{2(3\alpha+1)-3(2\alpha+1)\mu}{(\alpha+1)^{2}}$,we have$3(2 \alpha+1)(a_{3}-\mu a_{2}^{2})=3(b_{3}+\frac{1}{3}(x-2)b_{2}^{2})$$+ \beta(p_{2}+\frac{1}{4}(\beta x-2)p_{1}^{2})+\beta xp_{1}b_{2}$

.

(2.8)

Since rotations of also belong to, without loss of generality, we may assumethatis positive. Thus we now estimate.
$f$$Q^{\alpha}(\beta)$$a_{3}-\mu a_{2}^{2}$${\rm Re}(a_{3}-\mu a_{2}^{2})$, there exists $h(z)=1+k_{1}z+k_{2}z^{2}+\cdots$ $(|z|<1)$ with positive real
Sincepart, such that $g’(z)+zg’(z)=g’(z)h(z)$ . Hence, by equating coefficients, we getthat
and $4=(k_{2}+k_{1}^{2})/6$. So, by Lemma 2.1,
$g\in \mathrm{C}$$4=\mathrm{k}\mathrm{i}/2$$3{\rm Re}(b_{3}+ \frac{1}{3}(x-2)b_{2}^{2})=\frac{1}{2}{\rm Re}(k_{2}-\frac{1}{2}k_{1}^{2})$%$\frac{1}{4}(x+1){\rm Re} k_{1}^{2}$$\leq 1-\rho^{2}+(x+1)\rho^{2}\cos 2\phi$where$\mathrm{h}$$=k_{1}/2=\rho e:\emptyset$

for some$\rho$$(0\leq\rho \leq 1)$. We also have

(2.9)
,

7
THE FEKETE SZEGO PROBLEM$\beta{\rm Re}(p_{2}+\frac{1}{4}(\beta x-2)p_{1}^{2})=\beta{\rm Re}(p_{2}-\frac{1}{2}p_{1}^{2})+\frac{1}{4}\beta^{2}x{\rm Re} p_{1}^{2}$$\leq 2\beta(1-r^{2})+\beta^{2}xr^{2}\cos 2\theta$

where

3

$p_{1}=2re^{i\theta}$(2.10),

for some $r(0\leq r\leq 1)$ . Prom (2.8-10), we obtain$(2\mathrm{a}+1)(a\mathrm{a}-\mu a_{2}^{2})\leq 1-\rho^{2}+(x+1)\rho^{2}\cos 2\phi$$+2\beta(1-r^{2})+\beta^{\mathit{2}}xr^{2}\cos 2\theta+2\beta xr\rho\cos(\theta+\phi)$,(2.11)

and we now proceed to maximize the right-hand of (2.11). This function wil bewhenever all the parameters except are held constant.
denoted by$x$$\psi(x)$We consider first the case$\frac{2\alpha(1-\alpha)+2\beta(3\alpha+1)}{3(2\alpha+1)(1+\beta)}\leq\mu\leq\frac{2(3\alpha+1)}{3(2\alpha+1)}$

so that $0\leq x\leq 2/(1+\beta)$ . Since thewhen $t=x/(2-\beta x\cos 2\theta)$ , we have,$\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}-2t^{2}+t^{2}\beta x\cos 2\theta+2xt$$-2t^{2}+t^{2} \beta x\cos 2\theta+2xt\leq\frac{x^{2}}{2-\beta x\cos 2\theta}\leq\frac{x^{2}}{2-\beta x}$

is the largest.

Thus$\psi(x)\leq x+1+\beta(2+\frac{x^{2}}{2-\beta x})$$=1+2 \beta+\frac{2\{2(3\alpha+1)-3(2\alpha+1)\mu\}}{2(\alpha+1)^{2}-\beta\{(2(3\alpha+1)-3(2\alpha+1)\mu\}}$and with (2.11) this establishes the second inequality in the theorem. Equality occurs
only if$p_{1}= \frac{2x}{2-\beta x}=\frac{2\{2(3\alpha+1)-3(2\alpha+1)\mu\}}{2(\alpha+1)^{2}-\beta\{2(3\alpha+1)-3(2\alpha+1)\mu\}}$and the corresponding function$f$,$p_{2}=2$,$h$$=b_{3}=1$

is defined by$(f’(z))^{1-\alpha}((zf’(z))’)^{a}= \frac{1}{(1-z)^{2}}(\lambda\frac{1+z}{1-z}+(1-\lambda)\frac{1-z}{1+z})^{\beta}$

,

,

8
N. E. CHO AND S. OWA$\lambda=\frac{2(\alpha+1)^{2}+(1-\beta)(2(3\alpha+1)-3(2\alpha+1)\mu)}{4(\alpha+1)^{2}-2\beta(2(3\alpha+1)-3(2\alpha+1)\mu)}$.

We now prove the first inequality. Let$\mu\leq\frac{2\alpha(1-\alpha)+2\beta(3\alpha+1)}{3(2\alpha+1)(1+\beta)}$,so that $x\geq 2/(1+\beta)$ . With $x_{0}=2/(1+\beta)$ , we have$\psi(x)=\psi(x_{0})+(x-x_{0})(\rho^{2}\cos 2\phi+\beta^{2}r^{2}\cos 2\theta+2\beta\rho r \cos(\theta+\phi))$$\leq\psi(x_{0})+(x-x_{0})(1+\beta)^{2}$$\leq 1+\frac{(1+\beta)^{2}\{2(3\alpha+1)-3(2\alpha+1)\mu\}}{(\alpha+1)^{2}}$as required. Equality occurs only if $p_{1}=p_{2}=2$ ,function$f$$b_{2}=\mathrm{k}$$=1$, and the corresponding

is defined by$(f’(z))^{1-\alpha}((zf’(z))’)^{\alpha}= \frac{1}{(1-z)^{2}}(\frac{1+z}{1-z})^{\beta}$

Let $x_{1}=-2/(1+\beta)$ . We shall find that $\psi(x_{1})=1+2\beta$ , and the remaining
inequalities follow easily from this one. By an argument similar to the one above, we
obtain$\psi(x)\leq\psi(x_{1})+|x-x_{1}|(1+\beta)^{2}$.

$\leq-1+\frac{(1+\beta)^{2}\{3(2\alpha+1)\mu-2(3\alpha+1)\}}{(\alpha+1)^{2}}$if$x\leq x_{1}$, that is,$\mu\geq\frac{2(\alpha+1)^{2}+2(3\alpha+1)(1+\sqrt)}{3(2\alpha+1)(1+\beta)}$

Equality occurs only if $p_{1}=2i$ ,function is defined by$b_{2}=i$,$p_{2}=-2$,.

$b_{3}=-1$, and the corresponding$f$$(f’(z))^{1-\alpha}((zf’(z))’)^{\alpha}= \frac{1}{(1-iz)^{2}}(\frac{1+iz}{1-iz})^{\beta}$

Also, for$0\leq\lambda\leq 1$, we note that

9
THE FEKETE-SZEG\"O PROBLEM$\psi(\lambda x_{1})=\lambda\psi(x_{1})+(1-\lambda)\psi(0)$$\leq\lambda(1+2\beta)+(1-\lambda)(1+2\beta)=1+2\beta$,

so $\psi(x)\leq 1+2\beta$ for $x_{1}\leq x\leq 0$ , that is,$\frac{2(3\alpha+1)}{3(2\alpha+1)}\leq\mu\leq\frac{2(\alpha+1)^{2}+2(3\alpha+1)(1+\beta)}{3(2\alpha+1)(1+\beta)}$

Equality occurs only iffunction is defined by$p_{1}=\mathrm{b}$$=0$ ,$p_{2}=2$,$b_{3}=1/3$.

, and the corresponding$f$$(f’(z))^{1-\alpha}((zf’(z))’)^{\alpha}= \frac{1}{1-z^{2}}(\frac{1+z^{2}}{1-z^{2}})^{\beta}=\frac{(1+z^{2})^{\beta}}{(1-z^{2})^{1+\beta}}$

We now show that$\psi(x_{1})\leq 1+2/\mathit{3}$.

. Since$(-2+\beta x_{1}\cos 2\theta)t^{2}+2x_{1}t\rho \mathrm{c}\mathrm{o}\mathrm{e}(\theta+\phi)$$=(-2+ \beta x_{1}\cos 2\theta)\{t+\frac{x_{1}\rho\cos(\theta+\phi)}{-2+\sqrt x_{1}\cos 2\theta}\}^{2}+\frac{x_{1}^{2}\rho^{2}\cos^{2}(\theta+\phi)}{2-\beta x_{1}\cos 2\theta}$

for all real and$t$$2- \beta x_{1}\cos 2\theta=2+\frac{2\beta}{1+\beta}\cos 2\theta\geq 2-\frac{2\beta}{1+\beta}\geq 0$,

we have$\psi(x_{1})-(1+2\beta)\leq\rho^{2}(-1+(x_{1}+1)\cos 2\phi+\frac{\beta x_{1}^{2}(1+\mathrm{c}\mathrm{o}\mathrm{e}2(\theta+\phi))}{2(2-\beta x_{1}\cos 2\theta)})$.

Thus we consider the inequality.

$\beta x_{1}^{2}(1+\cos 2(\theta+\phi))+2(2-\beta x_{1}\cos 2\theta)(-1+(x_{1}+1)\cos 2\phi\leq 0$

After some simplifications, this becomes$4(\beta^{2}(\cos 2\phi+1)(\cos 2\phi-1)-\beta(1+\cos 2\theta+\sin 2\theta\sin 2\phi)-1-\cos 2\phi)\leq 0$,

which is true if.

$2\beta^{2}\cos^{2}\theta\sin^{2}\phi+2\beta\cos\theta\sin\theta\cos\phi\sin\phi+\cos^{2}\phi\geq 0$(2.12)

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N. E. CHO AND S. OWA

Now, for all real ,$t$$2t^{2}+2t\sin\theta\cos\phi+\cos^{2}\phi$ $\geq 0$

so, by taking $t=\beta$ c0s6)
theorem.

$\sin\phi$,, we obtain (2.12). This completes the proof of theRemark. Letting $\alpha=0$ in Theorem 2.3, we have the corresponding result obtained by London [14], which extend the earlier results by several authors [1,5,10-12].

For $\alpha=1$ in Theorem, we have the following
Corollary 2.1. Let $f\in Q^{1}(\beta)$ and be given by (Ll). Then for$1+(1+\beta)^{2}(8-9\mu)\overline{\overline{4}}$$1+ \mathit{2}\beta+\frac{2(8-9\mu)}{8-\beta(8-9\mu)}$$9|a_{3}-\mu a_{2}^{2}|\leq\{$$1+2\sqrt$$-1+ \frac{(1+\beta)^{2}(9\mu-8)}{4}$For each , there is a function in$\mu$$Q^{1}(\beta)$

if
if
if
if

$\mu\leq+_{91+\beta}^{8}$, we have,$\frac{8\beta}{\neg 9(1+\beta}\leq\mu\leq\frac{8}{\mathfrak{g}}$$\frac{8}{9}\leq\mu\leq\ovalbox{\tt\small REJECT}_{91+}^{82+\beta}$$\mu\geq\ovalbox{\tt\small REJECT}_{91+\beta}^{82+}$$\geq 0$$\beta$,,.

such that equality holds in all cases.

References

1. H. R. Abdel-Gawad and D. K. Thomas, The Fekete-Szego problem for strongly
close-tO-convex functions, Proc. Amer. Math. Soc, 114(1992), 345-349.
2. D. A. Brannan and W. E. Kirwan, On some classes of bounded $univalentfilnctio\eta$J. London Math. soc., $1(2)(1969)$ , 431-443.
3. C. Carath\’eodory, Uber den ariabilitatsbbereich der fourierschen konstanten vonpositiven harmonischen funktionen, Rend. Circ. Math. Palermo, 32(1911), 193217.
4. Y. M. Chiang, On a subclass of strongly gamma-starlike functions and quasiconformal extensions, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 45(1991), 21-27.
5. M. Darus and D. K. Thomas, -logarithinically convex functions, Indian J. Pure
Appl. Math., 29(10)(1998), 1049-1059.
6. P. L. Duren, Univalent functions, Springer-Verlag, New York, 1983.
7. M. Fait, J. G. Krzyz and J. Zygmunt, Explicit quasiconformal extensions for someclasses of univalent function, Comment. Math. Helv., 51(1976), 279-285.
8. M. Fekete and G. Szego, Eine Bermerkung uber ungerade schlichte function, J.
London Math. Soc, 8(1933), 85-89.
9. W. Kalplan, Uose-to-convex schlicht functions, Michigan Math. J., 1(1952), 169$\alpha$11
THE FEKETE-SZEGO PROBLEM

10. F. R. Keogh and E. P. Merkes, A coefficient inequality for certain classes of analytic
functions, Proc. Amer. Math. Soc, 20(1969), 8-12.
11. W. Koepf, On the Fekete-Szeg\"o problem for close-tO-convex functio ns, Proc. Amer.
Math. Soc, 101 (1987), 89-95.
12. W. Koepf, On the Fekete-Szego problem for close-tO-convex functions , Arch.
Math., 49(1987), 420-433.
13. Z. Lewandowski, S. S. Miller and E. Zlotkiewicz, Gamma-starlike function, Ann.
Univ. Mariae Curie-Sklodowska Sect. A, 28(1974), 53-56.
14. R. R. London, Fekete-Szego inequalities for close-tO-convex functions, Proc. Amer.
Math. Soc, 117(1993), 947-950.
15. K. I. Noor, On quasi-convex functions and related topics, Internat. J. Math. &Math. Sci., 10(1987), 234-258.
16. K. I. Noor and H. A. Alkhorasani, Proper ties of close-tO-conveing preserved bysome integral operators, J. Math. Anal. AppL, 112(1985), 509-516.
17. A. Pfluger, The Fekete-Szego inequality for complex parameters, Complex Variables
Theory AppL, $7(1986)$ , 149-160.
in the class S, Complex Variables Theory
18. A. Pfluger, On the functional
AppL, 10(1988), 83-95.
19. Ch. Pommerenke, Univalent functions, Vandenhoeck and Ruprecht, Gottingen,1975.
$|a_{3}-\lambda a_{2}^{2}|$20. J. Stankiewicz, Quelques proble mes extremaux dans les classes des fonctions
angulairement etoilees, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 20(1966),$\alpha-$59-75.

NAK EUN Cho, DEpARrMENT OF AppLIED MATHEMATIOS,
Pusan 608-737, KOREA (E-MAIL: NECHO@PKNU.AC.KR)$\mathrm{p}_{\mathrm{U}\mathrm{K}\mathrm{Y}\mathrm{O}\mathrm{N}\mathrm{G}}$NATIONAL UNIVERSITY,SHIGEYOSHI Owa, DEpARTMENT 0F MATHEMATICS, KINKI UNrVERSITY,-8502, JApAN (E-MAIL: OWA@MATH. KINDAI. .JP)$\mathrm{A}\mathrm{K}\mathrm{A}577$$\mathrm{A}\mathrm{C}$$\mathrm{H}\mathrm{I}\mathrm{G}\mathrm{A}\mathrm{S}\mathrm{H}\mathrm{I}\sim \mathrm{O}\mathrm{S}\mathrm{A}\mathrm{K}\mathrm{A}$,$\mathrm{O}\mathrm{s}\sim$