Generalized Distribution and Its Geometric Properties Associated with Univalent Functions

The purpose of the present paper is to introduce a generalized discrete probability distribution and obtain some results regarding moments, mean, variance, and moment generating function for this distribution. Further, we show that for specific values it reduces to various well-known distributions. Finally, we give a beautiful application of this distribution on certain analytic univalent functions.

The Application of Real Convolution for Analytically Evaluating Fermi-Dirac-Type and Bose-Einstein-Type Integrals

The Fermi-Dirac-type or Bose-Einstein-type integrals can be transformed into two convergent real-convolution integrals. The transformation simplifies the integration process and may ultimately produce a complete analytical solution without recourse to any mathematical approximations. The real-convolution integrals can either be directly integrated or be transformed into the Laplace Transform inversion integral in which case the full power of contour integration becomes available. Which method is employed is dependent upon the complexity of the real-convolution integral. A number of examples are introduced which will illustrate the efficacy of the analytical approach.

Certain Integral Operator Related to the Hurwitz–Lerch Zeta Function

The aim of the present paper is to investigate several third-order differential subordinations, differential superordination properties, and sandwich-type theorems of an integral operator Ws,bf(z) involving the Hurwitz–Lerch Zeta function. We make some applications of the operator Ws,bf(z) for meromorphic functions.

Fixed Point and Newton’s Methods in the Complex Plane

We revisit the necessary and sufficient conditions for linear and high order of convergence of fixed point and Newton’s methods in the complex plane. Schröder’s processes of the first and second kind are revisited and extended. Examples and numerical experiments are included.

Second Hankel Determinants for Some Subclasses of Biunivalent Functions Associated with Pseudo-Starlike Functions

We introduce second Hankel determinant of biunivalent analytic functions associated with λ-pseudo-starlike function in the open unit disc Δ subordinate to a starlike univalent function whose range is symmetric with respect to the real axis.

Geometric Properties of Cesàro Averaging Operators

Using positivity of trigonometric cosine and sine sums whose coefficients are generalization of Vietoris numbers, we find the conditions on coefficient {ak} to characterize the geometric properties of the corresponding analytic function f(z)=z+∑k=2∞akzk in the unit disc D. As an application, we also find geometric properties of generalized Cesàro-type polynomials.

Coefficient Bounds for Subclasses of Biunivalent Functions Associated with the Chebyshev Polynomials

We introduce and investigate new subclasses of biunivalent functions defined in the open unit disk, involving Sălăgean operator associated with Chebyshev polynomials. Furthermore, we find estimates of the first two coefficients of functions in these classes, making use of the Chebyshev polynomials. Also, we give Fekete-Szegö inequalities for these function classes. Several consequences of the results are also pointed out.

Uniqueness of the Sum of Points of the Period-Five Cycle of Quadratic Polynomials

It is well known that the sum of points of the period-five cycle of the quadratic polynomial fc(x)=x2+c is generally not one-valued. In this paper we will show that the sum of cycle points of the curves of period five is at most three-valued on a new coordinate plane and that this result is essentially the best possible. The method of our proof relies on a implementing Gröbner-bases and especially extension theory from the theory of polynomial algebra.

Connections between Certain Subclasses of Analytic Univalent Functions Based on Operators

In this paper, by applying the Hohlov linear operator, connections between the class SD(α), α≥0, and two subclasses of the class A of normalized analytic functions are established. Also an integral operator related to hypergeometric function is considered.