Fekete-Szeg\"{o} Problems for Certain Classes of Meromorphic Functions Using $q$-Derivative Operator
Received:April 26, 2017  Revised:March 23, 2018
Key Word: analytic function   meromorphic function   Fekete-Szeg\"{o} problem   $q$-derivative operator   $q$-Bessel function
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.\,11561001; 11271045), the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (Grant No.\,NJYT-18-A14), the Natural Science Foundation of Inner Mongolia (Grant Nos.\,2010MS0117; 2014MS0101; 2017MS0113), the Higher School Foundation of Inner Mongolia (Grant Nos.\,NJZY16251; NJZY17300; NJZY17301; NJZY18217) and the Natural Science Foundation of Chifeng.
 Author Name Affiliation Huo TANG School of Mathematics and Statistics, Chifeng University, Inner Mongolia $024000$, P. R. China H. M. ZAYED Department of Mathematics, Faculty of Science, Menofia University, Shebin Elkom 32511, Egypt A. O. MOSTAFA Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt M. K. AOUF Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Hits: 504
In this paper, we introduce two subclasses $\Sigma _{q}^{\ast }(\varphi )$ and\ $\Sigma _{q,\alpha }^{\ast }(\varphi )$ of meromorphic functions $f(z)$ for which $\frac{qzD_{q}f(z)}{f(z)}\prec \varphi (z)$ and \frac{-(1-\frac{\alpha }{q})qzD_{q}f(z)+\alpha qzD_{q}[zD_{q}f(z)] }{(1-\frac{\alpha }{q})f(z)-\alpha zD_{q}f(z)}\prec \varphi (z),\ \ \alpha \in \mathbb{C}\backslash (0,1],\ 0