数理解析研究所講究録
1062 巻 1998 年 84-88

84

UNIVALENCE OF CERTAIN
INTEGRAL. OPERATORS
Shigeyoshi OWA

Virgil PESCAR

30. 01.1998
Abstract

In this work some integral operators are studied and the authors determine conditions for the univalence of these integral operators.

1

INTRODUCTION

Let A be the class of the functions which are analytic in the unit disc
$U=\{z\in C_{i}|z|<1\}$ and $f(\mathrm{O})=f’(0)-1=0$ .
We denote by the class of the function $f\in A$ which are univalent inU.
Many authors studied the problem of integral operators which preserve
result is due to J. PfaltzgrafP [4].
the class S. In this sence an impor
a complex number andTHEOREM A [4]. If is univalent in, then the function$\mathrm{f}$$\mathrm{S}$’$\iota \mathrm{a}\mathrm{n}\mathrm{t}$$\mathrm{U},$$\mathrm{f}$$\alpha$$| \alpha[\leq\frac{1}{4}$(1)$G_{\alpha}(z)= \int_{l}^{*}[f’(\xi)]^{\circ}d\epsilon$

is univalent in U.

. If the function $g\in S$ andTHEOREM, then the function defined by$\mathrm{B}[3]$$\alpha$is a complex number,$| \alpha|\leq\frac{1}{4n}$(2)$G_{\alpha,n}(Z)= \int_{l}^{*}[_{\mathit{9}’} (u )]^{\alpha}du$is univalent in2$\mathrm{U}$for all positive integer$\mathrm{n}$.

PRELIMINARY RESULTS

We will need the following theorems in this work.
be a complex number,THEOREM$\mathrm{C}[2].\mathrm{L}\mathrm{e}\mathrm{t}\alpha$$f\in A$. If${\rm Re}\alpha>0$and85(3)$\frac{1-|_{Z}|^{2}R\mathrm{c}\alpha}{Re\alpha}|\frac{zf’’(Z)}{f(z)},|\leq 1$

for $a\mathrm{U}z\in U$ , then for any complex number$\beta,{\rm Re}\beta\geq$

Rea the function$(4)$F_{\beta}(z)=[ \beta\int_{\Phi}^{*}u^{\rho}-1f’(u)du]$is in the class S.
Ifan $\mathrm{d}|g(Z)|<1$
function is regular inTHEOREM$z\inU$$\xi\inU$,
andthe following inequalities holdthen for allin$\mathrm{D}[1]$.

$\mathrm{U}$$\mathrm{t}1\iota \mathrm{e}$$\mathrm{g}$$\mathrm{U}$(5)$| \frac{g(\xi)-g(z)}{1-\overline{g(z)}g(\xi)}|\leq|\frac{\xi-z}{1-\overline{z}\xi}|$and$|g’(z) \mathrm{I}\leq\frac{1-|g(_{Z})|^{2}}{1-|z|^{2}}$,(6)the equalities hold only in the case
REMARK [1]. For $z=0$ , from inequahty (5)$g(z)= \epsilon\frac{*+u}{1+\overline{\mathrm{u}}*}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}|\epsilon|=1$

and $|u|<1$(7)$| \frac{g(\xi)-g(0)}{1-\overline{g(0)}g(\epsilon)}|\leq|\xi|$and, hence$|g( \xi)\mathrm{I}\leq\frac{|\xi|+|g(0)|}{1+|g(\mathrm{o})||\epsilon|}$

.

.

(8)

Considering $g(\mathrm{O})=a$ and $\xi=z$ ,$|g(Z)| \leq\frac{|z|+|a|}{1+|a||X|}$

.

(9)

.

for all $z\in U$$g(\mathrm{O})=0$
function is regular inLEMMA SCHWARZ [1]. If
the following inequalities holdand [for all $z\in U,$$\mathrm{U},$$\mathrm{t}\mathrm{l}\iota \mathrm{e}$$\mathrm{g}$$g(\approx)|\leq 1^{\cdot}$$\mathrm{t}\mathrm{l}\iota \mathrm{e}\mathrm{n}$$|g(z)1\leq|z|$(10)

for all $z\in U$ , and $|g’(0)|\leq 1$ , the equalities (in inequality (10) for $z\neq 0$ ).
holdin the case $g(z)=\epsilon z$ , where$|\epsilon|=1$$\mathrm{o}n1\mathrm{y}$3MAIN RESULTS

THEOREM 1. Let
If$\alpha,\gamma$

be complex numbers, ${\rm Re}\alpha=a>0$ and $g\in A$$| \frac{g’’(Z)}{g’(z)}|\leq\frac{1}{n}$for$\mathrm{a}\mathrm{U}z\in U$an$\mathrm{d}$.

(11)

86

(12)

$| \gamma|\leq\frac{n+2a}{2}(\frac{n+2a}{n})^{T\epsilon}\hslash$then for any complex number$\beta,$${\rm Re}\beta\geq a$.

$c_{\beta_{7}.n}.(Z)= \{\beta\int_{0}$, the function(13)$u^{\beta}-1[g’(u)n]\gamma du\}^{i}$is in the class for all $n\in N$ $-\{1\}$ .
PROOF. Let us consider the function$\mathrm{S}$$f(z)= \int_{0}$.

$[g’(u^{n})]^{\gamma}du$.

(14)

The function,$p(z)= \frac{1}{|\gamma|}\frac{f’’(Z)}{f’(_{Z)}}$(15)

satisfies the inequahty (12), is regular in U.
where the constant
From (15) and (14) we obtain$|\gamma|$(16)$p(z)= \frac{\gamma}{|\gamma|}[\frac{nz^{h-1}J’t(z^{\hslash})}{g’(zn)}]$Using (16) and (11) we obtain(17)$|p(z)1<1$

for$\mathrm{a}\mathrm{U}z\in U$. For $z=0$ we have$p(\mathrm{O})=0$.

From (16) and Schwarz-Lemma it results$\mathrm{t}\mathrm{l}\iota \mathrm{a}\mathrm{t}$(18)$\frac{1}{|\gamma|}|\frac{f’’(_{Z)}}{f’(z)}|\leq|z|^{\mathrm{B}}-1\leq|z|$

for all $z\in U$ , and hence$( \frac{1-|z|^{2a}}{a})|\frac{zf’’(Z)}{f(z)},|\leq|\gamma|(\frac{1-|z|^{2\mathrm{n}}}{a})|z|^{\hslash}$

Let us consider. We have$\mathrm{Q}:[0,1]arrow R,$$Q(x)= \frac{(1-x^{2\sigma})}{}‘ x^{n}$

.

(19)

,$\mathrm{g}=|z|$$Q(x) \leq\frac{2}{n+2a}(\frac{n}{n+2a})^{\star_{\epsilon}}$for all$x\in \mathfrak{l}^{\mathrm{o},1}$]. From (20), (19) and (12) we obtain$( \frac{1-|z|^{2}\mathrm{B}}{a})|\frac{zf’’(z)}{f(z)},|\leq 1$

for$\mathrm{a}\mathrm{U}z\in U$$G_{\beta.\gamma}.,.$(20). Then, from (21) and Theorem, is in the class S.

$\mathrm{C}$it$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{S}$(21)

that the function87
THEOREM 2. Letfunction $g\in A$ ,$g(z)=z+a_{2}z^{2}+\ldots$$\alpha,$$\gamma$be complex numbers, Rea $=b>0$ and the. If(22)$| \frac{g’’(z)}{g’(_{Z)}}|<1$

for all $z\in U$ and the constantsatisfies the condition$|\gamma|$

1$|\gamma|\leq$(23)$\max_{1}\mathrm{I}*|\leq[\frac{1-|*1^{2b}}{b}1^{z}|_{1^{*+2}}+2{\rm Im}_{\mathrm{g}}\circ \mathrm{I}2|s|]$then for any complex number$\beta,$${\rm Re}\beta\geq b$

the function(24)$G_{\beta,\gamma}(z)= \{\beta\int_{\Phi}^{*}u^{\rho}-1[g^{J}(u)]^{\tau_{d\}^{*}}}u$

is in the class S.
Proof. Let us consider the function$f(z)= \int_{\mathit{0}}^{2}[g’(u)]^{\gamma}du$.

(25)

The function$h(z)= \frac{1}{|\gamma|}\frac{f’’(z)}{f’(z)}$

,

(26)

the constant [ satisfies the inequality (23), is regular in U.
From (26) and (25) we have$\mathrm{w}\mathrm{l}\iota \mathrm{e}\mathrm{r}\mathrm{e}$$\gamma|$$h(z)= \frac{\gamma}{|\gamma|}\frac{g’’(z)}{g’(_{Z)}}$

.

(27)

Using (27) and (22) we obtain$|h(z)|<1$ ,forand $|h(0)|=2|a_{2}|$ .
The above Remark applied to the function(28)$\mathrm{a}\mathrm{U}z\in U$$\mathrm{h}$

gives
(29)

$\frac{1}{|\gamma|}|\frac{f’’(z)}{f(z)},|\leq\frac{|z|+2|a_{2}|}{1+2|a_{2}||Z|}$for all $z\in U$ .
From (29) we obtain(30)$\frac{1-|z|^{2}b}{b}|\frac{zf’’(Z)}{f(z)},|\leq|\gamma|\frac{1-|Z|^{2b}}{b}[z|\frac{|z|+2|a_{2}|}{1+2|a_{2}||z|}$

for all $z\in U$ . Hence, we$1\iota \mathrm{a}\mathrm{v}\mathrm{e}$$\frac{1-|z\mathrm{I}^{2b}}{b}|\frac{zf’’(z)}{f’(z)}|\leq|\gamma|\max_{1}1*\mathrm{I}\leq[\frac{1-|z|2b}{b}|_{Z}|\frac{|z|+2|a_{2}|}{1+2\mathrm{I}a_{2}|\mathrm{I}z|}]$.

(31)

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From (31) and (23) we obtain(32)$\frac{1-|_{X}|^{2b}}{b}|\frac{zf’’(Z)}{f(z)},.|\leq 1$

for all $z\in U$ . From Theoremby (24) is in the class S.

$\mathrm{C}$, it follows tllat tlte function$G_{\beta.\gamma}$

definedReferences[1] Z. Nehari, Conformal mapping, Mc Graw-Hill Book Comp., New York,1952 (Dover. Publ. Inc., 1975)[2] N.N. Pascu, An improvement of Becker’s univalence crilerion, Proceedings of the Commemorative Session Simion Stoilow,,(1987), 43-48.
$\mathrm{B}\mathrm{r}\mathrm{a}_{S^{\mathrm{O}}}\mathrm{v}$[3] N.N. Pascu, V. Pescar, On
Pfaltzgra fF, Studia
32, 2(1990), 185-192.

$\mathrm{t}\mathrm{l}\iota \mathrm{e}$integral operators of Kim-Merkes and, Univ. Babe5-Bo1yai, Cluj-Napoca,$(\mathrm{M}\mathrm{a}\mathrm{t}1_{1}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{C}\mathrm{a})$[4] J. Pfaltzgraff, Univalence of the integral
Math. Soc. 7(1975), No. 3, 254-256.

$\int_{0}[f’(t)1\mathrm{c}d${, Bull. London[5] C. Pommerenke, Univalent functions,Gottingen, 1975.

”Transilvania” University ofFaculty of Science$\mathrm{B}\mathrm{r}\mathrm{a}_{5^{\mathrm{O}\mathrm{V}}}$

Department of$\mathrm{M}\mathrm{a}\mathrm{t}1_{1\mathrm{e}\mathrm{m}}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{C}}\mathrm{s}$2200$\mathrm{B}\mathrm{r}\mathrm{a}_{5}\mathrm{o}\mathrm{V}$ROMANIA

Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577-8502
Japan