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The class $\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$ defined in the domain $|z|<1$ satisfying \begin{align*} {\rm Re\,} e^{i\phi} \Big(\dfrac{}{}(1\!-\!\alpha\!+\!2\gamma)\!\left({f}/{z}\right)^\delta +\Big(\alpha\!-\!3\gamma+\gamma\Big[\dfrac{}{}\left(1-{1}/{\delta}\right)\left({zf'}/{f}\right) \\ +{1}/{\delta}\left(1+{zf''}/{f'}\Big)\right]\Big) \left.\dfrac{}{}\left({f}/{z}\right)^\delta \!\left({zf'}/{f}\right)-\beta\right)>0, \end{align*} with the conditions $\alpha\geq 0$, $\beta<1$, $\gamma\geq 0$, $\delta>0$ and $\phi\in\mathbb{R}$ generalizes a particular case of the largest subclass of univalent functions, namely the class of Bazilevi\v c functions. Moreover, for $0<\delta\leq\frac{1}{(1-\zeta)}$, $0\leq\zeta<1$, the class $\mathcal{C}_\delta(\zeta)$ be the subclass of normalized analytic functions such that \begin{align*} {\rm Re}{\,}\left(1/\delta\left(1+zf''/f'\right)+(1-1/\delta)\left({zf'}/{f}\right)\right)>\zeta,\quad |z|<1.
\end{align*} In the present work, the sufficient conditions on $\lambda(t)$ are investigated, so that the non-linear integral transform \begin{align*} V_{\lambda}^\delta(f)(z)= \left(\int_0^1 \lambda(t) \left({f(tz)}/{t}\right)^\delta dt\right)^{1/\delta},\quad |z|<1, \end{align*} carries the functions from $\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$ into $\mathcal{C}_\delta(\zeta)$. Several interesting applications are provided for special choices of $\lambda(t)$. These results are useful in the attempt to generalize the two most important extremal problems in this direction using duality techniques and provide scope for further research.
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참고문헌 (17)
참고문헌 신청
R. M. Ali / 2012 / Starlikeness of integral transforms and duality / J. Math. Anal. Appl 385(2):808~822
R. M. Ali / 2013 / Convexity of integral transforms and duality / Complex Var. Elliptic Equ 58(11):1569~1590
R. M. Ali / 1995 / Convexity and starlikeness of functions defined by a class of integral operators / Complex Variables Theory Appl 26(4):299~309
B. C. Carlson / 1984 / Starlike and prestarlike hypergeometric functions / SIAM J. Math. Anal 15(4):737~745
J. H. Choi / 2002 / Geometric properties of convolution operators defined by Gaussian hypergeometric functions / Integral Transforms Spec. Funct 13(2):117~130
R. M. Ali / 2013 / Convexity of integral transforms and duality / Complex Var. Elliptic Equ 58(11):1569~1590
R. M. Ali / 1995 / Convexity and starlikeness of functions defined by a class of integral operators / Complex Variables Theory Appl 26(4):299~309
B. C. Carlson / 1984 / Starlike and prestarlike hypergeometric functions / SIAM J. Math. Anal 15(4):737~745
J. H. Choi / 2002 / Geometric properties of convolution operators defined by Gaussian hypergeometric functions / Integral Transforms Spec. Funct 13(2):117~130
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