Application of convolution theory on non-linear integral operators
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Abstract
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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References
[1] R. M. Ali, A. O. Badghaish, V. Ravichandran and A. Swaminathan, Starlikeness of integral transforms and duality, J. Math. Anal. Appl. 385 (2012), no. 2, 808– 822. Google Scholar
[2] R. M. Ali, M. M. Nargesi and V. Ravichandran, Convexity of integral transforms and duality, Complex Var. Elliptic Equ. 58 (2013), no. 11, 1569–1590. Google Scholar
[3] R. M. Ali and V. Singh, Convexity and starlikeness of functions defined by a class of integral operators, Complex Variables Theory Appl. 26 (1995), no. 4, 299–309. Google Scholar
[4] B. C. Carlson and D. B. Shaffer, Starlike and prestarlike hypergeometric func- tions, SIAM J. Math. Anal. 15 (1984), no. 4, 737–745. Google Scholar
[5] J. H. Choi, Y. C. Kim and M. Saigo, Geometric properties of convolution oper- ators defined by Gaussian hypergeometric functions, Integral Transforms Spec. Funct. 13 (2002), no. 2, 117–130. Google Scholar
[6] S. Devi and A. Swaminathan, Integral transforms of functions to be in a class of analytic functions using duality techniques, J. Complex Anal. 2014, Art. ID 473069, 10 pp. Google Scholar
[7] S. Devi and A. Swaminathan, Starlikeness of the Generalized Integral Transform using Duality Techniques, 24 pages, Submitted for publication, (http://arxiv.org/abs/1411.5217). Google Scholar
[8] P. L. Duren, Univalent functions, Grundlehren der Mathematischen Wis- senschaften, 259, Springer, New York, 1983. Google Scholar
[9] A. Ebadian, R. Aghalary and S. Shams, Application of duality techniques to starlikeness of weighted integral transforms, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 2, 275–285. Google Scholar
[10] R. Fournier and S. Ruscheweyh, On two extremal problems related to univalent functions, Rocky Mountain J. Math. 24 (1994), no. 2, 529–538. Google Scholar
[11] Y. C. Kim and F. Rønning, Integral transforms of certain subclasses of analytic functions, J. Math. Anal. Appl. 258 (2001), no. 2, 466–489. Google Scholar
[12] Y. Komatu, On analytic prolongation of a family of operators, Mathematica (Cluj) 32(55) (1990), no. 2, 141–145. Google Scholar
[13] P. T. Mocanu, Une propri et e de convexit e g en eralis ee dans la th eorie de la repr esentation conforme, Mathematica (Cluj) 11 (34) (1969), 127-133. Google Scholar
[14] R. Omar, S. A. Halim and R. W. Ibrahim, Convexity and Starlikeness of certain integral transforms using duality, Pre-print. Google Scholar
[15] S. Ruscheweyh, Convolutions in geometric function theory, S eminaire de Math ematiques Sup erieures, 83, Presses Univ. Montr eal, Montreal, QC, 1982. Google Scholar
[16] R. Singh, On Bazileviˇc functions, Proc. Amer. Math. Soc. 38 (1973), 261–271. Google Scholar
[17] S. Verma, S. Gupta and S. Singh, Order of convexity of integral transforms and duality, Acta Univ. Apulensis Math. Inform. No. 38 (2014), 279–296. Google Scholar