CERTAIN CLASSES OF UNIVALENT AND MEROMORPHIC FUNCTIONS INVOLVING Q-ANALOGUE OPERATORS

Abstract

This research explores some generalised derivative and integral operators associated with quantum calculus in the space of normalised analytic and bi-univalent functions in the open unit disc and meromorphic functions in the punctured unit disc. Some ba- sic definitions and initial results of the geometric function theory used in this study are stated. These operators are defined by using the convolution technique and quantum cal- culus. In this current work, generalised derivative operators involving the q-analogue of the Opoola operator are constructed to introduce new classes of analytic and meromor- phic functions. These classes are defined as associated with q-calculus. Some analytical and geometrical properties of the classes are studied, including coefficient inequalities, growth bounds, distortion theorems, extreme points, and closure theorems. A new con- cept of q-neighbourhoods, radii of q-starlikeness, and q-convexity are defined. Fur- thermore, the upper bounds for the Fekete-Szegö problem are solved. The coefficient bounds, convolution properties, and the q-Jackson’s integral representations for these subclasses are also studied. Subsequently, new subclasses of meromorphic functions using a new derivative operator involving the q-Opoola operator are introduced. Fur- thermore, new subcategories of bi-univalent functions linked to orthogonal polynomials, namely Gegenbauer, Faber, and Euler polynomials on the open unit disc are derived us- ing the subordination principle. The main focus here is to compute the first two-term coefficients of the aforementioned subcategories, and later to obtain the Fekete- Szegö inequality. In addition, some of the classical properties, including sufficient conditions, the upper and lower of its partial sums, and some results based on convolutions are de- rived. Finally, harmonic univalent functions and harmonic meromorphic functions are studied. New classes of the Borel distribution-type Mittag-Leffler function for univalent harmonic functions and the q-Al-Oboudi operator for meromorphic harmonic functions are given. Their properties are studied. The findings of the research generalised previ- ous results with the new-found methods and latest styles in geometric function theory.