On certain subclasses of
analytic
functions
involving
alinear
operator
群馬工業高等専門学校
斎藤
斉
HITOSHI
SAITOH
Department
of
Mathematics,
Gunma National
College
of
Ibchnology
Tbribamachi
580, Maebashi,
Gunma
371\cdot 8530, Japan
January 25,
2005
Abstract
Acertain linear
operator
defined
by
aHadamard
product
or
convolution
for fimctions
$\mathrm{w}\overline{\mathrm{h}}.\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r},\mathrm{e}$analytic
in
the open
unit disk
is
introduced.
The
object
of the
present
paper
is
to
derive
some
properties of this
linear
operator.
84
4
$\mathrm{I}_{\cap}\mathrm{f}_{\Gamma\theta}^{-}d.\downarrow\iota c\mathrm{b}’on$ $/\Delta_{\backslash p}$:
de
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$k_{n\mathrm{c}},\#_{l\grave{\mathit{0}}\}1\mathrm{S}}ot*hex_{\mu w1}$
$f(\#)=$
$z^{\ell}+ \sum_{n\simeq/}^{\mu}a_{\alpha\uparrow\ell}\mathrm{z}^{nrp}$
$(f^{\mathrm{e}N})$
$(\sqrt\{f)$
$\vee\swarrow \mathit{1}_{1\mathfrak{l}\mathrm{C}}^{\mathrm{t}}\mathrm{A}\mathrm{o}n_{\sim}$
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:
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A
$l\sim-\mathrm{A}$
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we.
de
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$(f* \theta)(\mathrm{Z})=\sum o\circ a_{n}l_{\eta}Z^{h}$
$n=0$
$L_{\iota}\mathrm{f}$
$\oint_{r}(\mathrm{q} \mathrm{c}\}\mathcal{B})--\sum_{h=0}^{\Phi}\frac{(\alpha)_{4|}}{(_{-}\mathrm{c})_{\alpha r}}g^{\gamma_{l}*p}$
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2)
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.
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$(f_{\downarrow}\mathit{3})$$\wedge$
$\mu_{\mathrm{n}C}k_{\acute{\eta}\eta}f_{f\mathrm{g}})\in A_{r}\prime^{\backslash }\leq sa_{\mathfrak{l}}^{\mathrm{t}}d\mathrm{I}\iota\iota_{e}/ent/-\beta^{\sim\vee\aleph}\nearrow k_{V}l^{\mathrm{t}},\mathrm{k}_{\mathrm{Q}}$
$\Leftrightarrow$
$\beta_{\mathrm{e}}\{^{\backslash }f\frac{zf’\mathrm{r}\epsilon)}{l\neq)}\}p_{0}$$(z\epsilon\cup)$
$(f_{\backslash }\mathit{1}_{\mathrm{A}})$$f_{\mathrm{t}} \oint_{p}r-- wlen\mathrm{t}l\nearrow$
CofiVRX
$<^{-}-\Rightarrow$
$l$$(/|S^{-}\mathrm{I}$
$TJ_{1l\mathit{5}\mathrm{C}}$ $\sigma \mathrm{u}b\mathrm{c}l\alpha\sigma \mathrm{s}\epsilon \mathrm{s}$
$\mathrm{q}re_{-}$
$dena\mathrm{f}\mathrm{e}\mathit{4}$
$\mathrm{b}\nearrow$
$\mathrm{S}_{\ell}^{*}a_{\nu\iota}d$
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’
$\chi_{p}(\mathcal{V}+r, l)f/S)=.\cdot\frac{\mathrm{z}^{P}}{(/-\mathrm{Z}\mathrm{J}^{V\#}}*p_{S\mathit{1}=D^{\vee\uparrow p\dashv}f(S)}/$
$(.f_{1}\mathrm{A})$
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$\mathrm{X}_{\rho}(v\mathrm{r}p_{2}\mathcal{V}tp\dagger f)fl8)=\frac{\mathcal{V}\dagger\ell}{E^{V}}\mathrm{J}_{\mathit{0}}^{8}t^{\gamma\sim f}f(\star$
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$\neq[\not\leq)\epsilon- \mathrm{A}p/$
$Wl_{-}h_{V\mathrm{e}}$
$F(x_{\ell}(a,\mathrm{c})f(\mathrm{g}))^{/}=\alpha*[a*l,$
$\mathrm{c}\mathit{1}ffZ\mathrm{I}-(a-p)4(a, \mathrm{c})\beta?)\nearrow$
$(\mathit{2}_{\backslash }\mathrm{f})$$.w/|\mathrm{e}\mathrm{r}e$
,
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le
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86
$a>\mathit{0}-$
If
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$| \leqq\frac{\{}{\mathrm{A}},\frac{1\S 1}{|-|\mathrm{P}|}$
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$\mathrm{F}l\#)=$
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$\mathrm{F}\{(\not\in)\grave{/}\mathrm{S} ana\mathit{1}y^{\mathrm{f}_{l}\iota}/t\}\backslash \mathrm{U}$
$\emptyset \mathrm{n}d$ $/\mathrm{F}$$\mathrm{F}l\Xi)=2\mathrm{p}(\mathrm{t})$
$\sqrt \mathrm{t}_{p\Gamma \mathrm{C}}$$p\ell^{l\mathrm{z})}\}^{1}\mathrm{S}.c_{\mathfrak{j}}r|$
For
$su\mathrm{c}l_{\backslash }$ $+_{\mathrm{L}1l1\mathrm{c}\mathrm{b}_{\acute{g}-\nu\prime}\mathrm{s}}$ $\mathrm{r}\mathrm{v}\mathrm{L}l_{\lfloor\alpha \mathrm{v}\mathrm{e}}$$($
1
$\gamma^{l}(8)|\leqq$
$|\sim|8|^{\iota}$
$\tau(\mathrm{g})/<-f$
$/F(\mathit{0})\simeq \mathit{0}$
$\eta_{\mathrm{W}\mathrm{S}}$$\}$
a
$/$$y\mathrm{t}_{\mathfrak{l}’\mathrm{C}}/r$
$\backslash \bigcup_{an}\mathrm{A}$
}
1
$\Psi^{lE)l_{=}^{\angle}}f$
$\mathrm{b}_{\nearrow}$
$L_{\mathrm{h}0[]\nu\iota\lambda}\mathit{2}|2$
.
)
$\mathrm{A}_{\iota rw\iota \mathrm{q}\mathit{2},f}$
‘
$=\underline{1}$
$=$
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$\mathrm{T}\mathrm{t}_{\iota \mathrm{e}\mathrm{r}\mathrm{e}}\mathrm{t}_{\Gamma 4})$ ’ $\mathrm{W}\mathrm{C}$ $\}_{\backslash a\mathrm{v}\mathrm{c}}$ $|$$-\uparrow$
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.
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.
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en
$f(S)\grave{\mathrm{I}}S$
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$\mathrm{P}\# l$$<\mathrm{P}^{toof>}$
$S_{/\acute{n}_{-}ce}$
$h(\ell,p)pg)--f\mathrm{r}z)\ovalbox{\tt\small REJECT}$
$*lftl, \ell)\oint \mathit{1}8)=\not\in f_{l\mathrm{S})}’/\ell’|w\not\subset \mathrm{A}_{4Y\mathrm{e}}$
$| \mathrm{Z}rightarrow^{(}t\not\geq)-fl2)\mathrm{P}|\leqq\frac{|\yen|}{1-|\not\in|}$
$|_{\mathrm{I}}^{\mathrm{Q}}|\backslash \nearrow$$\rangle 0$ $\grave{|}\eta$ $1 \mathrm{Z}|<\frac{\mathrm{P}}{\mathrm{P}+1}$
Q.E.D.
$\mathrm{I}+$
$\mu_{\mathrm{Z})-}-2+\mathrm{q}_{L}z^{\mathrm{Z}}+-\wedge\cdot$
$\iota \mathrm{s}\backslash$$\alpha \mathfrak{n}\alpha 1- \mathrm{b}_{\mathrm{C}}^{\backslash }\gamma$
$|$ $\grave{1}v\iota 1]/\mathrm{f}\mathrm{l}_{Q\mathrm{r}_{1}}$
$\mathrm{f}[\mathrm{g})1^{\backslash }\mathrm{S}\sigma \mathrm{b}_{\mathrm{r}}1_{1\mathrm{k}\mathrm{e}}’\prime \mathrm{I}\eta\backslash \backslash \mathrm{g}|<\frac{1}{\mathrm{Z}}$
$\lfloor_{4}\not\subset$ $\mathrm{f}(8)\in \mathrm{A}_{\mathrm{P}}$
$\mathrm{I}+|\frac{\S 1(8)}{PZ^{\rho\sim/}}-1|<1$
$(\neq\Leftarrow \mathrm{u})$
,
$\mathrm{P}^{-\mathrm{v}a1_{\theta \mathrm{v}\backslash }\{[f}\mathrm{c}\mathrm{r}\mathfrak{n}\mathrm{v}\epsilon\chi\backslash 1\mathrm{f}1$ $\backslash \S\backslash <\overline{\mathrm{P}}\star\overline{1}\mathrm{L}$
$<praa.\cdot f>\mathrm{S}_{\mathrm{t}_{\mathrm{b}\mathrm{C}\mathrm{A}}^{\backslash }}$
$-\lambda \mathfrak{n}.A$
$\lambda_{\uparrow}\iota\uparrow-\uparrow \mathrm{x},\mathfrak{p})$
$we_{-}$
.
$\}_{1\mathrm{A}\mathrm{v}\iota}$ $|$
$(1-+ )- \mathrm{P}|\leqq\frac{|\S|}{1\sim \mathfrak{l}\mathrm{Z}\backslash }$
$J$ $\}_{\backslash }\mathrm{R}\cdot/$$1+\mathrm{B}_{\mathrm{R}}$
$\succ o$
$|\mathfrak{n}\backslash$$\backslash \mathrm{s}\backslash <\frac{\mathrm{P}}{\mathrm{P}*1}$
$.\Theta,$
$\mathrm{E},$$D$
,
CDVo
$\mathrm{t}\mathrm{t}\mathrm{o}\mathfrak{j}\psi\sqrt/2_{\iota}^{[+}$.
$\mathrm{C}\mathrm{M}_{c\iota \mathrm{c}\Re_{8^{\mathrm{d}^{-}}}}$)
I
$\rho$ $\mathrm{f}18$)
$=\mathrm{z}+\mathrm{o}_{\mathrm{z}}\mathrm{z}_{\dagger}^{2_{-}}--$.
$\mathfrak{l}\mathrm{S}\backslash \alpha \mathrm{v}\iota_{\nearrow}\mathrm{Q}\backslash \mathrm{b}^{\mathrm{t}}\mathrm{c}_{-}$ $\wedge\wedge \mathrm{A}$
sO
$(\mathrm{t}_{\acute{1}5}\mathrm{t}^{\iota}\mathrm{e}\mathrm{s} |\mathrm{f}^{\mathrm{I}_{18)-}}\{|<1 \backslash \mathrm{A}\backslash 1\mathrm{J})$
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$\grave{|}\mathfrak{n}$$|\mathrm{S}|<_{2}[perp]$
$’.\mathrm{t}\}_{\backslash \mathrm{P}\mathrm{f}\mathrm{t}}$
88
$<\dot{f}\mathrm{r}\circ\circ \mathrm{f}>$ $\sqrt\epsilon|^{?\mathrm{t}\mathrm{A}}\mathrm{t}$
-
$\cdot 1=\mathrm{z}^{(\mathrm{r}^{lf)}}$
$\mathrm{T}^{(_{\backslash \rho_{\mathrm{Y}\backslash }}}$
$(\mathrm{r}\downarrow \mathrm{g})$
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,
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$(2_{\backslash }+)$
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$\grave{\iota}\eta \mathrm{U}J$ $+\mathrm{k}_{Bt4}\mathrm{f}\mathrm{i}_{\mathrm{Y}4}$R
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2
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$\alpha\sim(w*()(z\backslash \dagger[\alpha\dashv)[\mathrm{Z}|^{\mathrm{Z}}$
$\Theta.E.P_{-}$
$\mathrm{R}_{\mathrm{t}}\{\frac{t_{\mathfrak{p}^{(\mathrm{h}\mathrm{S}[,\mathrm{C})\oint 12)}}}{\theta \mathrm{C}^{\mathrm{c}_{\mathrm{t}’}\mathrm{c})\mathrm{G}^{1\mathrm{S})}}}\}>0$ $\mathrm{k}’[8|<\frac{\mathrm{Z}\nwarrow}{2\mathrm{a}\dagger \mathrm{I}*\frac{\mathrm{g}\mathrm{q}\star \mathfrak{l}}{}}$
$\frac{\mathrm{G}_{\mathrm{W}}\mathfrak{l}1\alpha \mathrm{v}\mathrm{v}2_{\backslash }\mathrm{S}}{/}$
.
$\mathrm{L}_{\mathrm{e}_{-}}\mathrm{t}$
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)
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$\}|\not\in)\mathrm{e}hd$
\S 1\S )
$\mathrm{e}\mathrm{S}^{*}$I\S
$| \frac{\mathrm{f}\not\in)}{7^{\not\in \mathrm{J}}}-1|<[$ $+1b)\backslash \mathrm{t}\mathrm{S}$$\sigma \mathrm{b}_{r}\backslash \mathrm{i}\mathrm{k}\not\subset$
$\grave{|}\eta$
$| \mathrm{g}|<\frac{1}{3}-$
$\mathrm{s}_{\backslash }+$ $\mathrm{B}\mathrm{a}(\{+\frac{\mathrm{s}3^{||}(\mathrm{z})}{\Im^{1}\mathrm{t}\not\in)}\}>-\{(\not\in\epsilon \mathrm{u})$
is
$7^{1_{l\mathrm{V}\backslash }\{\backslash _{\mathrm{y}}}-\mathrm{y}\mathrm{q},\mathrm{C}^{0\mathfrak{n}\mathrm{v}\mathrm{e}x}$ $\grave{1}\mathrm{h}$$\mathrm{L}_{Q}\lambda$ $\mathrm{f}^{-[\mathrm{z})\mathrm{e}\mathrm{A}}$
’
$\mathrm{t}\downarrow z$)
$\epsilon$
A
$\mathrm{S}_{1+}$.
$\mathrm{B}_{R}\{\{-+\frac{\not\in\S^{1\prime}\beta)}{\int^{1}(\not\in)}\}>rightarrow|$$(8 \epsilon\cup)$
{
$|<_{\mathrm{L}}|$
$(\epsilon\epsilon \mathrm{u})’+\mathfrak{t}_{\iota\Omega \mathfrak{n}}$ $\S\downarrow \mathrm{a})$Ss
$\mathrm{C}o\mathrm{Y}\mathrm{t}\vee \mathrm{C}\mathrm{X}\grave{\mathrm{I}}\vee$)
$| \mathrm{g}\{<\frac{9-\sqrt{|7}}{\mathrm{k}}$
$fl^{g})\epsilon A_{t}an\mathrm{A}$
$l^{(l)\epsilon,4_{P}}s_{\theta}\not\in_{l\mathrm{S}}\backslash \mathrm{f}^{\mathrm{I}},ps$$)> \frac{\alpha_{-}}{\mathrm{R}+\mathrm{C}}$
$(E\epsilon U)/\mathcal{N}here$
$\mathit{0}\geq fud\mathrm{C}_{\sim}^{2}f$
If
At
$\frac{\mathrm{C}_{\mathit{0}\ulcorner \mathit{0}}11\alpha \mathrm{r}\mathrm{y}\mathit{2}_{\backslash }?}{/-}$
.
$\llcorner_{\rho}\lambda f[Z)\epsilon \mathrm{A}_{\Gamma}$
$\emptyset nA\mathit{1}^{l\mathrm{Z})\epsilon}s_{r}^{*}(\frac{\ell}{2}),$
$’.,e_{\nearrow}\backslash$$\hslash e[\frac{\mathrm{z}f^{l_{(8)}}}{\#^{l8)}}\}\gamma\frac{p}{2}$
$(\dot{\mathrm{z}}\epsilon \mathrm{U})$
.
If-
$| \frac{\mathrm{f}\mathrm{i}\mathrm{z})}{8^{(\mathrm{S})}}-1|<$1
$(\not\leq\epsilon U)\prime \mathrm{f}\mathrm{i}_{Ir1}\theta^{[\mathrm{S})\grave{|}\mathrm{S}}r^{-V^{g}\mathit{1}entf_{\gamma}},\sigma.\mathrm{f}_{\mathit{0}\Gamma}l,\backslash k\mathrm{e}\grave{/}\eta$
$|\mathrm{Z}|<(\sqrt{\ell^{\mathrm{a}}+lr+\mathrm{f}}-(p\# l))/2$
$(\mathrm{M}_{\mathit{0}\mathrm{C}}\mathrm{C}_{\mathrm{T}}ref^{W})$ $L_{\mathrm{C}}\tau$ $\#(\mathrm{S})\epsilon$
A
$a\nu\uparrow\circ \mathfrak{l}$$f^{(3)6}s^{\gamma}( \frac{/}{\mathit{2}})$
If
$\mathfrak{l}8\overline{\in}U)_{/}+\})\mathrm{e}\eta+l?)|\mathrm{s}\backslash \sigma \mathrm{h}_{\mathrm{K}}1_{\grave{\mathrm{I}}}\mathrm{k}\mathrm{e}$ $\grave{\mathfrak{l}}\cap$
$|\mathrm{P}|<\sqrt{\mathit{2}_{-}}-\uparrow$
$\mathrm{s}_{\iota\downarrow rP^{\theta \mathit{5}e}}u_{a}\mathrm{t}$ $/_{\mathrm{I}}l3)\simeq(+\mathrm{C}_{\{}?+\mathrm{C}_{\iota}\mathrm{Z}^{\mathrm{a}}+--\cdot/^{1}\mathrm{S}\eta\eta\iota\uparrow\mu^{1}\mathrm{c}a_{\mathfrak{U}}\mathrm{o}\mathrm{L}$
$\Sigma a+_{\grave{1}\sigma}f_{\mathrm{T}C\mathrm{S}}’$
$\mathrm{R}e\{\}_{1}(3)\}>\mathit{0}$
$\int_{\eta}U\sigma$
$\mathrm{T}h_{e1\}$we
$h_{qV8}$
$=|\mathrm{k}^{l}(\mathrm{Z})|\leqq$
$\frac{2\mathrm{k}\mathrm{e}\{\}_{1}(8)\}}{l-|\mathrm{E}|^{2}-}$.
$l_{\vee}\mathrm{e}_{\sim}\mathrm{m}\mathrm{m}\mathrm{A}2.\mathit{3}$ $;\mathrm{s}\backslash$$l+_{4\grave{t}\mathfrak{l}1}d^{\mathit{1}}$
$l|\nearrow$ $\alpha s\grave{|}f?\theta-$ $\angle_{Cr\mathrm{n}W\mathrm{L}\mathrm{q}}$.
$\mathit{2}.\mathit{2}.|$$A_{p\ell}/\mathcal{X}/nl$
$\iota_{\epsilon t?\mathrm{t}mq}\mathit{2}_{\iota}f$
so
$e\mathrm{f}:l^{l\not\in)\in h}$
$\sigma_{a}\mathrm{A}_{5}\prime \mathrm{A}^{1}\mathrm{e}s$
$U/$
$wl_{1}et^{r}4$
$\mathit{0}\underline{z}\mathit{1}$and
$C\underline{\mathit{2}}f$in
$| \not\in|<\frac{a(\sqrt{a^{2}+z\mathrm{c}*l}-\iota)}{\alpha^{2_{-}}+2\mathrm{C}}$
$\mathrm{I}\oint$$>\mathit{0};_{l1}U’$
’
$tl_{(\mathrm{f}\mathit{7}1}$
We
$l_{1\mathrm{f}\mathrm{l}^{\mathrm{y}\rho}}$$\}>\mathit{0}$
$\grave{l}n$ $|3|<. \frac{a(\sqrt{a^{\mathrm{a}}+2\mathrm{C}\dagger \mathfrak{l}}-/)}{a^{\mathit{1}}+2\mathrm{C}}$$/a\eta A$
$\mathrm{R}\not\in\{\frac{L(\mathrm{q} l,\mathrm{C})\mathrm{f}(g)}{*(\alpha,\iota)\oint 3)}\}\nearrow 0$
L
$e\lambda^{-}ftS$
)
$\not\in A_{P}$
$\alpha\prime_{1}d$$\uparrow lS$
$\}>o$
$/\eta U,$
$\tau l_{|p\eta_{1}}$$\kappa_{\mathrm{e}},f$
$)\in \mathrm{s}_{\rho}^{*}(_{\check{\mathrm{z}}}^{\triangle})$
$f_{\frac{\mathit{1}_{(\mathrm{f}\mathrm{f})}}{\prime l8J}l>\mathit{0}}?.\grave{l}/1(g$
}
$< \frac{p}{P+\mathrm{Z}}$
’
$\ovalbox{\tt\small REJECT} \mathrm{a}_{\nu \mathrm{L}}p\mathrm{L}mfl\mathrm{Z})\grave{\mathrm{I}}\mathrm{S}$
$r^{-\vee ale\mathrm{r}\mathrm{t}/}\nearrow$
$s\mathrm{b}_{V}l,\backslash \mathrm{k}\mathrm{e}\}_{1\mathrm{I}}$
$| \mathrm{S}|<\frac{\ell}{\rho_{+2}}$
$l_{-eX}f(Zy\epsilon A$
$n_{\mathit{0}}d$ $.\beta^{lZ)\mathit{6}}S^{\gamma(_{2}^{\mathit{1}})}$$lf$
fle
$[ \frac{k)}{?l\mathcal{Z})}\}>\mathit{0}//\uparrow\backslash U$
,
$>0$
$\prime^{\mathrm{l}}\mathrm{n}$I
$\mathrm{S}l<\frac{/}{\grave{\mathrm{J}}}J\ovalbox{\tt\small REJECT} f_{l\mathrm{Z})}lS\backslash \sigma \mathrm{f}_{\mathcal{E}|V}\mathit{1},\prime k\acute{\mathfrak{c}}$
$’\eta\backslash$ $f \Sigma/<’\frac{f}{\mathit{3}}$
$ae_{1\nu 1}d$
$.\mathrm{B}\not\subset\{$$\theta^{\mathfrak{l}5)\in}\kappa_{f}(\frac{\ell^{\mathrm{z}}}{2p\#/})$
..
$1+ \frac{\yen \mathrm{f}^{\prime_{\mathit{1}}}(\mathrm{g})}{\mathrm{F}’ t?)}\}p-\{$$| \mathrm{g}|<\frac{\sqrt{7}-f}{\mathit{3}}$
$s_{\Gamma\nearrow}^{*}wl_{\ell \mathrm{I}’}e$
$\mathit{1}\mathrm{r}/Z/<’t_{0}J$
$\underline{\mathrm{B}\mathrm{a}\mathfrak{l}\mathfrak{n}\alpha \mathrm{r}- \mathrm{k}.\mathit{2}.2}$
.
$\ovalbox{\tt\small REJECT}(\mathrm{Z})=\frac{\mathrm{A}}{z^{\alpha+}}\int_{\mathit{0}}^{3}r^{a_{f^{-\int}}}f_{lk)}$
d
え
$(S_{lL}((|\eta))$
$c_{w\llcorner}d$
$\mathrm{f}^{i_{(S)=\chi}}(a/a+\int)f(\Sigma)\epsilon S^{*}w/\zeta\oint \mathrm{l}$
$l^{1}t1/\mathrm{z}|<r_{or}$
$\sqrt l_{\mathrm{C}t\epsilon_{-}}$$(\mathrm{A}>Z)$
$(a–z)$
$L_{f}r\neq(\not\leq)\in A$
$a\gamma \mathrm{t}\mathrm{A}$$dX\epsilon S^{*}$
$w/\mathrm{E}$
$\mathrm{C}\epsilon-/\wedge[$$\mathrm{T}\}_{|\mathrm{e}\circ\uparrow}$
$f_{l\mathrm{Z})}$ $\mathrm{I}^{\backslash }S$
stqr
$l_{\grave{\mathrm{f}}}\mathrm{k}\mathrm{c}$$l\mathrm{h}\backslash$ $|\mathrm{B}$
}
$<\nabla_{by}$
$\mathrm{v}\prime he\nu\not\in$
$\{$
$\mathrm{r}_{\mathrm{o}}^{\backslash }rightarrow-\frac{\sim \mathrm{z}_{\dagger}\sqrt{3+\mathrm{C}^{1}}}{\mathrm{C}-1}$
$\zeta \mathrm{C}>1)$
$\mathrm{P}_{0}^{\wedge}--\frac{1}{?_{-}}$
$\mathrm{t}\mathrm{c}=\sqrt)$
$L_{\mathrm{C}}tf\mathrm{r}\mathrm{z}^{-})\epsilon,\mathrm{A}$
4
$\mathrm{n}\theta[$ $\ovalbox{\tt\small REJECT}_{l\not\in)=\frac{2}{\mathbb{Z}}\int_{0}^{2}f(A-)d\lambda}\epsilon \mathrm{S}^{*}\nearrow$ $\sigma$$S+_{a\}^{\wedge}/;\mathrm{k}\rho}$
92
$\backslash \mathit{3}$
$S_{\mathit{0}}m\mathrm{c}a\ell pl\mathfrak{l}’\mathrm{C}a- b^{1}\mathit{0}|\mathit{0}\mathrm{S}o$
{
$d^{\backslash }\mathrm{I}hf’\ell l|t_{l\grave{\mathit{0}}}t\mathit{5}^{(\Lambda}.b\theta \mathrm{r}d\mathfrak{i}na\mathrm{b}\backslash \grave{\mathit{0}}\mathrm{r}]$
$. \frac{\mathit{0}pf,\mathit{3}.\mathfrak{l}}{\vee}$
.
$\mathrm{L}_{\iota}t\oint \mathrm{g}$
)
$\hslash r\mathrm{L}\mathit{0}/ff^{(E)}\iota_{cana}/_{\gamma}+_{l^{\mathrm{t}}G\grave{l}\eta}lJ$
.
$\mathcal{T}\mathit{4}_{en}\mathrm{v}\swarrow e\sigma_{f}at/l\mathrm{q}t$
$f(\not\in 2t\grave{S}\mathrm{s}_{1A}\mathrm{b}o\mathrm{r}\mathrm{A}’|\cap\alpha \mathrm{f}\not\subset\tau_{\phi}\beta lZ)(\backslash W\Gamma^{\mathrm{t}}/t\mathrm{f}_{\ell t\eta}p_{\mathrm{Z}j\prec?}(ff))$
if
$ff^{lS)_{/5\ell r;\}\acute{/}\psi\alpha}’/_{en}t}\nearrow$
$f( \mathit{0})-arrow\oint(\mathit{0})a_{\Lambda}\iota d_{-}f(U)\subset fflU)$
.
.
$\gamma\epsilon C,mA$
$\beta ef\beta lll)\theta f>c$
扁
$.\mathrm{g}_{\mathrm{e}\prime}-\underline{\mathit{3}.\mathit{2}}$
.
$L_{\mathrm{e}X}$$\kappa_{\rho}\backslash (\mathrm{G}{}_{r}\mathrm{C}jc1er1\mathit{0}\mathrm{f}e+l_{\iota}ec_{-f_{\theta \mathit{5}\mathit{5}}}a_{7}C\hslash nct\neg^{l}\mathit{0}\kappa \mathrm{s}f\epsilon \mathrm{A}_{f^{2}}$
$\sigma\{44da+ \frac{z(Xp(4,\mathrm{C})p_{s))’}}{P\dot{*}\mathrm{r}a,\mathrm{c})flZ)}\prec \mathrm{h}\}\not\leq)$
$\mathrm{r}z\epsilon Uj\nearrow$
$w^{(_{\{\mathrm{e}v^{-}e}}\}_{1}(S)\grave{/}S\mathrm{c}_{\mathit{0}}$
nvex
$/n\backslash []\mathcal{N}/\mathrm{f}:hl_{1}\zeta \mathit{0})\vee-(a\prime dP_{\backslash }pf\mathrm{h}(s)\}_{P\mathit{0}_{r}}$
$(a\underline{\mathit{2}}r)$
$l_{\ n\mathrm{t}a}\mathit{2}$
.
$f,$
$Wph_{\mathit{0}\psi e}$
$\mathrm{T}h_{1^{\backslash }\mathrm{S}}$
$\gamma i\eta e\lambda nSd\alpha t$
$’ \oint\backslash$$\cdot f\mathrm{r}\mathrm{z}$
.
)
$\epsilon\ (a+f_{J}\zeta)$
’
$\epsilon h_{e\prime}n$
$\mathrm{F}\mathrm{t}x\iota n$
$Mm4\mathit{3},$
$lJ$
$\grave{(}++_{0}/l\mathrm{o}ws\mathrm{t}h\emptyset t$
$\mathrm{A}_{t}$$\mathit{0}\underline{\geq}pr$
$\oint(Z)\prec\}_{\mathrm{t}}$
(ff)
$-ner^{F}efi_{\vee L}/$
$\prec/)[\Sigma)$
$wl_{1l}\backslash c\mathrm{A}\}\gamma\uparrow eansfi\mathrm{e})\in \mathrm{R}_{f}(a,\mathrm{c})$
$h\ulcorner$
a
$||$
A2
?
$\Theta.\mathrm{E}.D$
.
$\frac{\mathrm{D}_{p}f.\mathit{3}.\mathit{3}}{-}$
.
$\mathrm{f}^{\mathrm{j}}(\not\equiv)=\frac{\gamma_{+}p}{\mathcal{Z}^{\mathrm{Y}}}\int_{\theta}^{\mathrm{E}}\lambda^{l-l}f- \mathrm{t}T)d\mathrm{f}$
$(f(F)\in A_{r})$
$\underline{\neg 7_{\Phi\vdash 49\eta 31}\mathit{2}.}$
$fi\mathrm{s})\in\beta_{p^{\zeta \mathrm{q},\mathrm{C})}}--\geq \mathrm{F}(8)\in \mathrm{R}_{f}(0,\mathrm{C})$
$<r^{\gamma \mathit{0}of>\mathrm{F}\mathrm{r}b\nu \mathrm{n}}..\cdot\dot{\mathrm{q}}|loVQ_{-}\circ[_{q}fi_{r\mathrm{I}\prime}^{\mathrm{t}\backslash }k^{\backslash },\sigma\eta,$
$WC_{-}\mathit{1}_{\iota a\vee \mathrm{e}}$
$.\mathrm{z}\ovalbox{\tt\small REJECT}_{1B)+}’\gamma \mathrm{F}(z)=(7+\mathrm{p})\# lE)$
$\backslash \mathcal{N}Q_{-}\mathrm{w}\mathrm{r}^{\backslash }’ \mathrm{k}$
$\lambda_{\mathrm{P}}\mathrm{t}\mathrm{q},$
$\zeta_{-})\theta^{(\S)=}\phi_{pl^{\mathrm{q}}},\mathrm{c})*f_{lS})=(\phi_{\ell}*t)\mathrm{r}\mathrm{z})$
So
w2
$\}_{\mathrm{t}\mathrm{A}V^{\mathrm{Z}}}$$\varphi_{p}*(z\mathrm{F}’)(\not\in)+l(\phi_{\mathrm{P}}*\mathrm{F}).\mathrm{t}\cdot z)=[\gamma_{\dagger}p)(\oint_{r}*f)(g)$
$\mathrm{L}l_{5\iota^{\backslash }}nff\mathrm{f}he_{\vee}k\mathrm{c}t$
$\mathrm{Z}(\phi_{f}*\mathrm{F})/l\mathrm{S})=(\phi_{\Gamma}*\not\in \mathrm{F}’)(\mathrm{Z})$
$\iota_{\vee\nu e}$
$/|a\nu e_{-}$
$\mathrm{g}(\psi_{\rho}*\mathrm{F})’(3)+\mathrm{Y}(\phi_{\mathrm{F}}*\mathrm{F})\mathrm{t}\mathrm{E})=$
$( \mathrm{Y}\uparrow P.)(\oint_{P}*\mathrm{f})(\mathrm{g})$
Lek
$\mathfrak{p}l\mathrm{Z})=p\cdot(\phi_{P}*\mathrm{F})(\mathrm{g})$
TAem
$\mathrm{w}eo\mathrm{b}+_{\overline{a}\iota^{1}’\iota}$
84
/1
$lZ$
)
$w\mathrm{h}_{\grave{|}}\mathrm{c}\mathrm{k}\mathfrak{l}\mathrm{m}\backslash [_{1}r’\not\subset s$ $+h_{\lambda}\mathrm{t}$
$\ell^{(\mathrm{Z})}\prec(_{1}(2)$
$t_{y}$
$L_{\xi_{\hslash t\prime}}\mathrm{r}\wedge \mathit{9}_{\mathrm{f}}f$.
$\tau/_{1ef\varphi}keJ$
$we_{-}\}_{\iota a\mathrm{v}\not\subset}$$\ovalbox{\tt\small REJECT}\{8$
)
$6p_{f}\backslash (a, \mathrm{c})$
(.
(Q.
$\mathrm{E},$$D$
.
$)$
$\Rightarrow$
$\mathrm{F}(\mathrm{E})\in S^{*}$
$\Rightarrow$
$\mathrm{F}^{(\mathrm{g})}\in \mathrm{K}$
0
$\# 4\eta c,b^{f}ons$
$\prec \mathrm{A}lS)$
$\mathrm{r}\mathrm{z}\in\cup)\mathrm{r}$
$\underline{\mathrm{T}h_{\mathrm{d}o\mathcal{R}\mathrm{o}\mathrm{n}3.3}}$
.
$\neq l\mathrm{P}$)
$\epsilon R_{f}$
ta,
c)
$(\emptyset^{\underline{2}}\ell, d>\mathit{0})$
$<pr\theta af\rangle$
\lfloor
之え
$\mathrm{F}_{hn\uparrow}$$\mathrm{L}pm’\gamma[]\triangleleft 2_{\backslash }f\prime w\not\subset\}_{1\mathrm{d}Ve}$
a
$\chi_{-p}(\propto+1, \mathrm{C})f\ell \mathrm{z})=(p.\cdot\ell \mathfrak{t}\mathrm{s})+(a-p))*(n, \mathrm{c}\mathrm{J}fl\mathrm{Z})$
$\mathrm{T}^{-}\alpha kj\eta\theta$$p_{lSj}$
$S_{\mathrm{I}f|d_{-}}^{\backslash }$
$f\mathrm{t}\#)\mathrm{e}\mathrm{R}_{\rho}^{d}(\mathrm{A}, \mathrm{C})/$
$lt\backslash$$4./l_{o\mathrm{W}\mathrm{S}}\mathrm{t}\mathrm{h}_{\mathrm{Q}}\mathrm{t}$
$\backslash ?\mathrm{I}^{\aleph}(\alpha {}_{r}\mathrm{C})(\S)=\frac{\alpha\S \mathrm{P}^{1}1\S)}{\mathfrak{p}.\mathrm{P}^{(\S\rangle\backslash \mathfrak{g}\triangleleft}}.+\alpha\gamma(8)+(_{1}-*]\mathrm{P}\square \S)$
$\prec \mathrm{k}(\mathrm{F})$
,
$\star.\backslash _{\backslash ^{\mathrm{Q}_{\iota}}}+$
$\mathrm{I}_{\mathrm{p}}^{\psi}(a, \mathrm{c})(\mathrm{g})=\frac{\mathrm{z}\mathrm{t}’\mathfrak{t}l)}{\mathrm{L}_{{}_{u}\mathrm{P}(\mathrm{g})+_{\overline{\aleph}}^{\underline{\mathrm{Q}}-\mathrm{L}}}}.+\mathrm{P}^{(8)}\prec$
A
12)
P
化い 火一一。
3.1
/
$\mathrm{w}4$
夏
$\alpha \mathrm{v}\not\subset$.
$\mathfrak{p}(\not\in)\prec \mathrm{k}l2)/$
怯辻
$\grave{1}\mathrm{S}$$\prec \mathrm{t}_{\backslash }\mathrm{t}8)$
$’\vee/(\backslash |\backslash (\mathrm{c} \mathrm{b}\backslash \mathfrak{W}^{\eta}\mathrm{S} \}(*)\in \mathrm{R}_{\mathrm{P}}\iota\alpha, \mathrm{c})$
$\Theta$.E.
$D$
,
$(\mathrm{M}_{\grave{1}}1|_{\mathrm{e}\nu_{\nearrow}^{\wedge}}\mathrm{M}_{\mathrm{o}\mathrm{c}\mathrm{a}\mathfrak{n}\iota\lambda}\mathrm{M} \mathrm{R}_{4\mathrm{A}}\mathrm{A}_{\mathrm{C}})$
$\mathrm{A}1\mathrm{t}$ $\alpha\sim \mathrm{c}ot1\mathrm{v}\epsilon \mathrm{x}$ $+_{1\Lambda^{\mathrm{Y}1\mathrm{C}_{-}}}+_{1\grave{\mathrm{O}}\mathrm{h}\mathrm{S}}$
a
$r\epsilon_{-}$$5\mathrm{b}_{r}|_{\mathfrak{l}\mathrm{k}\mathrm{c}}^{\backslash }$
$\mathrm{R}_{\gamma^{\aleph}}(\alpha_{/}\mathrm{C})\mathrm{C}$
$\mathrm{R}_{\mathrm{f}}^{\beta}$(A.C)
$(\alpha>\beta\geq 0)$
$<\ell r\mathit{0}\theta f>[\#$
$?^{-0}-$
)
$+\mathrm{k}\grave{\downarrow}\mathrm{s}+\}_{1e\mathrm{o}\mathrm{r}\iota\backslash \cdot \mathrm{A}}$
$.\mathrm{v}\mathfrak{n}\mathrm{g}a\mathrm{n}_{\mathrm{S}}$ $\mathrm{T}\mathfrak{b}e\circ r\mathrm{e}_{\mathrm{b}\{\mathrm{A}}3_{\iota}3$
He
$\mathfrak{n}\alpha$ $\mathrm{W}$[
(kssum2
$\iota_{\mathrm{P}}^{0(}\mathrm{t}\alpha\prime \mathrm{t}_{-})\mathrm{t}*)$$\mathrm{L}_{R}*\mathrm{S}_{0}\mathrm{b}_{4}$
$\alpha\}-\mathrm{b}_{\grave{\mathrm{t}}}+\mathrm{r}_{\wedge\nu}\mathrm{p}_{\mathfrak{g}\backslash \mathrm{y}\backslash }\mathrm{t}\backslash \grave{\mathrm{t}}\mathfrak{n}\mathrm{u}\nearrow \mathrm{t}\mathrm{h}_{e\mathrm{n}}\mathrm{I}_{\mathrm{f}}^{u}.(\alpha,\mathrm{c})\downarrow 8_{\theta}\rangle$
’
$\mathrm{k}\mathrm{t}.\mathrm{U}$).
$\mathrm{F}_{t\mathrm{o}\mathrm{w}\backslash }\eta_{ut\epsilon w\backslash }3_{\backslash }.3_{J}.$
.
$*(\mathrm{s})\epsilon \mathrm{R}_{\mathrm{P}}$to,
$\mathrm{c}$
)
$\nearrow\{\mathrm{t}_{\lambda}+-\backslash \backslash \mathrm{s}$$\prec\backslash \mathrm{r}_{1^{1S)}}$
$\mathrm{T}\}_{\mathfrak{i}\grave{1}\mathrm{s}}\grave{\iota}\mathrm{v}_{\wedge r^{1\grave{\mathrm{t}}Q}\mathrm{s}}$
$\mathrm{g}_{\mathit{0}}(x_{\mathfrak{p}}(o_{1/}\mathrm{t})\mathrm{f}\{8_{b}))’\in|_{\gamma(\mathrm{U}1}$
$p\cdot*\mathrm{C}\circ\iota,\mathrm{c})\mathrm{f}^{\mathrm{t}\mathrm{p}_{\mathrm{o}})}$$,\mathrm{A}1_{\mathrm{S}\mathit{0}}$
$1_{\mathrm{P}}^{\beta}( \alpha,\sigma_{\vee})\mathrm{t}\S)=(\backslash -\frac{\beta}{d}).\frac{g(\mathrm{x}_{\mathrm{P}}\iota \mathrm{q}_{l}\mathrm{C})\mathrm{I}(\not\in\cdot))}{\ell*(\mathrm{q},\mathrm{c})\mathrm{f}^{\{\mathrm{s})}}+\frac{\mathrm{P}}{\alpha}\mathrm{I}_{\mathrm{r}}^{a}(\alpha,\mathrm{c})(\mathrm{z})l$
$S_{\backslash ^{\backslash }}\mathfrak{n}\mathrm{t}\not\subset-\alpha \mathrm{k}_{<\{}\alpha \mathfrak{n}\mathrm{A}\}_{\backslash (\mathrm{U}),\mathrm{f}\mathrm{i}}\backslash \backslash \mathrm{s}\mathrm{C}0\mathrm{Y}\backslash \mathrm{v}\mathrm{e}\mathrm{x}.+)\mathrm{h}_{\mathrm{e}r\mathrm{p}}\mu_{4}\mathrm{I}_{?}^{\beta}(\propto,\iota)(\mathrm{g}_{0})\epsilon \mathrm{k}(\cup)$
$\dot{\mathrm{I}}\mathrm{f}$
:
$\backslash \iota_{\mathrm{o}\mathrm{v}t\mathrm{S}}$
℃
$\ltimes$、
$\lambda \mathrm{t}$ $\mathfrak{h}^{\mathrm{P}}\mathrm{t}\lambda,$$\mathrm{C}$)
$\mathrm{t}8$)
$\prec\wedge \mathrm{h}(\mathrm{g})$ $\tau \mathfrak{t}_{1\alpha \mathrm{t}^{-}}|^{\backslash }5\prime 4$
(\S )
$\epsilon \mathfrak{p}_{\mathrm{P}}^{\beta}(\mathrm{q},\mathrm{C})\epsilon$ $\mathrm{W}\grave{\iota}\mathrm{s}w\backslash \mathrm{Q}p_{4}\mathfrak{n}\mathrm{s}$ $\mathrm{R}_{\mathrm{p}}^{\aleph}(\mathrm{A}, \mathrm{C})$ $\mathrm{C}$ $\mathrm{R}_{\mathrm{p}}^{\mathrm{p}}[l\lambda, \mathrm{C})$ $\otimes_{!}$E.
$D$
,
se
Reference
$\mathrm{s}$l.S.D.Bernardi,
Convex and starhke univalent
functions,
Trans. Amer.
Math.
Soc.
135(1969),
$429\cdot 446$
.
$2.\mathrm{B}.\mathrm{C}.\mathrm{C}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{s}\mathrm{o}\mathrm{n}$
and
D.B.Shaffer,
Stau.e
and prestarLke
$\mathrm{A}ypergeome\mathrm{f}\dot{\mathrm{f}\mathrm{i}c}$functions,
SLAM
J.
Math.
Anal.
15(1984),
$737\cdot 745$
.
$3.\mathrm{P}.\mathrm{E}\mathrm{e}\mathrm{n}\mathrm{i}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{u}\mathrm{r}\mathrm{g}$
,
S.S.Miner,
P.T.Mocanu
and
M.O.Reade,
On 8Briot-Bouquest
ae.fferentiaJ subordin
a
tion,
Rev.
Roumaine Math.
$\mathrm{P}\mathrm{u}\mathrm{r}\mathrm{e}\epsilon$Appl.
29(1984),
567
$\cdot 573$
.
$4.\mathrm{R}.\mathrm{J}.\mathrm{L}\mathrm{i}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{a}$,
SOme classes of regular univalent
knctions,
Proc. Amer.
Math.
Soc.
16(1965),
$755\cdot 758$
.
$5.\mathrm{T}.\mathrm{H}.\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{G}\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{o}\mathrm{r},$
mctions
whose derivative has apositive zeal part
,
Trans.
Amer.
Math. Soc.
104(1962),
$532\cdot 537$
.
$6.\mathrm{T}.\mathrm{H}.\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{G}\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{o}\mathrm{r}$
,
The radius ofunivalence ofcertain
analytic
functions
$I$
,
Proc. Amer.
Math.
Soc.
14(1963),
$514\cdot 520$
.
$7.\mathrm{S}.\mathrm{S}.\mathrm{M}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{r}$
,
P.T.Mocanu and
M.O.Reade,
All
$a\cdot cotlvex$
mctions
are univalent and
starlike,
Proc. Amer. Math.
Soc.
37(2)
(1973),
$553\cdot 554$
.
8.S.Ruscheweyh,
New
criteria
for univalent
functions,
Proc. Amer. Math. Soc.
49(1975),
$109\cdot 115$
.
$9.\mathrm{H}.\mathrm{S}\mathrm{a}\mathrm{i}\mathrm{t}\mathrm{o}\mathrm{h}$
