Geometric properties of analytic functions generated by q-analog operators associated with special functions

Abstract

This thesis investigates various classes of integral and derivative operators generated by special functions associated with q-calculus for analytic univalent and p-valent functions. The study employs geometric function theory and uses a convolution method to define these operators. Additionally, convolution operators associated with a q-Mittag- Leffler function of univalent and p-valent functions are constructed. Several geometric properties of analytic functions are derived. These include coefficient inequalities, distortion and growth bounds, extreme points, integral mean inequality, subordinate factor sequence, partial sums, and radius theorems. The study also provides results on differential subordination, including best dominant and best subordinate theorems and sandwich-type theorems, by utilizing a subordination class defined by a q-derivative operator with the q-exponential function. Fekete-Szegö inequality of bi-univalent functions involving the q-exponential function using (m;n)-Lucas polynomials is also determined. Furthermore, the q-Hurwitz Lerch Zeta function is used to define a new qanalog integral operator and introduces a class of analytic non-Bazilevi˘c functions. This class is used to get subordination results. In addition, Hankel and Toeplitz’s determinants are also studied. Moreover, a general (p;q)-Bernardi integral operator for p-valent functions is introduced and subclasses of g-uniformly q-starlike and q-convex p-valent functions of order h are defined using this operator. The study also investigates several subordination consequences, such as containment relationships, Hadamard products, and integral-preserving property. Finally, comprehensive geometric properties for harmonic p-valent functions involving the q-Mittag-Leffler function operator are examined, and consequences for a family of harmonic (p;q)-starlike functions associated with the general (p;q)-Bernardi integral operator are derived.