A certain subclass of univalent meromorphic functions defined by a linear operator associated with the Hurwitz-Lerch zeta function

In this paper, we study a linear operator related to Hurwitz-Lerch zeta function and hypergeometric function in the punctured unit disk. A certain subclass of meromorphically univalent functions associated with the above operator defined by the concept of subordination is also introduced, and its characteristic properties are studied

Certain subclasses of analytic functions defined by a new general linear operator

Hypergeometric functions are of special interests among the complex analysts especially in looking at the properties and criteria of univalent. Hypergeometric functions have been around since 1900’s and have special applications according to their own needs. Recently, we had an opportunity to study on q-hypergeometric functions and quite interesting to see the behavior of the functions in the complex plane. There are many different versions by addition of parameters and choosing suitable variables in order to impose new set of q-hypergeometric functions. The aim of this paper is to study and introduce a new convolution operator of q-hypergeometric typed. Further, we consider certain subclasses of starlike functions of complex order. We derive some geometric properties like, coefficient bounds, distortion results, extreme points and the Fekete-Szego inequality for these subclasses

Hypergeometric Inequalities for Certain Unified Classes of Multivalent Harmonic Functions

In this paper, we consider unified classes PH (m, A,B) H and QH (m, A,B) H of multivalent harmonic functions F = H +G∈H(m) . Some hypergeometric inequalities for the functions of the class H(m) defined by generalized hypergeometric functions to be in these unified classes and its sub classes TPH (m, A,B) H and TQH (m, A,B) H , respectively, are obtained. Results, involving some integral operators are also given. Further, some special cases of the results are mentioned

Inequalities of harmonic univalent functions with connections of hypergeometric functions

Let SH be the class of functions f = h + (g) over bar that are harmonic univalent and sense-preserving in the open unit disk U = {z : vertical bar z vertical bar < 1} for which f(0) = f'(0) - 1 = 0. In this paper, we introduce and study a subclass H(alpha, beta)of the class SH and the subclass NH(alpha, beta) with negative coefficients. We obtain basic results involving sufficient coefficient conditions for a function in the subclass H(alpha, beta) and we show that these conditions are also necessary for negative coefficients, distortion bounds, extreme points, convolution and convex combinations. In this paper an attempt has also been made to discuss some results that uncover some of the connections of hypergeometric functions with a subclass of harmonic univalent functions

Application of Mittag–Leffler function on certain subclasses of analytic functions

The purpose of the present paper is to find the sufficient conditions for some subclasses of analytic functions associated with Mittag-Leffler function to be in subclasses of k−ST[A, B] and k−UCV[A, B] involving Jack’s Lemma. Further, we discuss certain inclusion results and characteristic properties of an integral operator related to MittagLeffler function.Publisher's Versio

Geometrical Theory of Analytic Functions

The book contains papers published in the Mathematics Special Issue, entitled "Geometrical Theory of Analytic Functions". Fifteen papers devoted to the study concerning complex-valued functions of one variable present new outcomes related to special classes of univalent functions, differential equations in view of geometric function theory, quantum calculus and its applications in geometric function theory, operators and special functions associated with differential subordination and superordination theories and starlikeness, and convexity criteria

Geometric Properties of Generalized Hypergeometric Functions

In this article, Using Hadamard product for $_4F_3\left(^{a_1,\, a_2,\, a_3,\, a_4}_{b_1,\, b_2,\, b_3};z\right)$ hypergeometric function with normalized analytic functions in the open unit disc, an operator $\mathcal{I}^{a_1,a_2,a_3,a_4}_{b_1,b_2,b_3}(f)(z)$ is introduced. Geometric properties of $_4F_3\left(^{a_1,\, a_2,\, a_3,\, a_4}_{b_1,\, b_2,\, b_3};z\right)$ hypergeometric functions are discussed for various subclasses of univalent functions. Also, we consider an operator $\mathcal{I}^{ a,\frac{b}{4},\frac{b+1}{4},\frac{b+2}{4},\frac{b+3}{4} }_{ \frac{c}{4}, \frac{c+1}{4}, \frac{c+2}{4},\frac{c+3}{4} }(f)(z)$= z\, _5F_4\left(^{a,\frac{b}{4},\frac{b+1}{4},\frac{b+2}{4},\frac{b+3}{4}}_{\frac{c}{4}, \frac{c+1}{4}, \frac{c+2}{4},\frac{c+3}{4}}; z\right)*f(z), where, $_5F_4(z)$ hypergeometric function and the $*$ is usual Hadamard product. In the main results, conditions are determined on $a,b,$ and $c$ such that the function z\, _5F_4\left(^{a,\frac{b}{4},\frac{b+1}{4},\frac{b+2}{4},\frac{b+3}{4}}_{\frac{c}{4}, \frac{c+1}{4}, \frac{c+2}{4},\frac{c+3}{4}}; z\right) is in the each of the classes $\mathcal{S}^{*}_{\lambda}$, $\mathcal{C}_{\lambda}$, $UCV$ and $\mathcal{S}_p$. Subsequently, conditions on $a,\,b,\,c,\, \lambda,$ and $\beta$ are determined using the integral operator such that functions belonging to $\mathcal{R}(\beta)$ and $\mathcal{S}$ are mapped onto each of the classes $\mathcal{S}^*_\lambda$, $\mathcal{C}_{\lambda}$, $UCV$, and $\mathcal{S}_p$.Comment: 44 Pages. arXiv admin note: text overlap with arXiv:2205.1338

Convolution Properties for Certain Classes of Analytic Functions Defined by q

We investigate convolution properties and coefficients estimates for two classes of analytic functions involving the q-derivative operator defined in the open unit disc. Some of our results improve previously known results

Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations

For a wide variety of quantum potentials, including the textbook `instanton' examples of the periodic cosine and symmetric double-well potentials, the perturbative data coming from fluctuations about the vacuum saddle encodes all non-perturbative data in all higher non-perturbative sectors. Here we unify these examples in geometric terms, arguing that the all-orders quantum action determines the all-orders quantum dual action for quantum spectral problems associated with a classical genus one elliptic curve. Furthermore, for a special class of genus one potentials this relation is particularly simple: this class includes the cubic oscillator, symmetric double-well, symmetric degenerate triple-well, and periodic cosine potential. These are related to the Chebyshev potentials, which are in turn related to certain ${\mathcal N}=2$ supersymmetric quantum field theories, to mirror maps for hypersurfaces in projective spaces, and also to topological $c=3$ Landau-Ginzburg models and `special geometry'. These systems inherit a natural modular structure corresponding to Ramanujan's theory of elliptic functions in alternative bases, which is especially important for the quantization. Insights from supersymmetric quantum field theory suggest similar structures for more complicated potentials, corresponding to higher genus. Our approach is very elementary, using basic classical geometry combined with all-orders WKB.Comment: 50 pages, 3 figure

Taming the cosmological constant in 2D causal quantum gravity with topology change

As shown in previous work, there is a well-defined nonperturbative gravitational path integral including an explicit sum over topologies in the setting of Causal Dynamical Triangulations in two dimensions. In this paper we derive a complete analytical solution of the quantum continuum dynamics of this model, obtained uniquely by means of a double-scaling limit. We show that the presence of infinitesimal wormholes leads to a decrease in the effective cosmological constant, reminiscent of the suppression mechanism considered by Coleman and others in the four-dimensional Euclidean path integral. Remarkably, in the continuum limit we obtain a finite spacetime density of microscopic wormholes without assuming fundamental discreteness. This shows that one can in principle make sense of a gravitational path integral which includes a sum over topologies, provided suitable causality restrictions are imposed on the path integral histories.Comment: 19 pages, 4 figures. Comments on general covariance added. To be published in Nucl. Phys.