PARTIAL SUMS OF GENERALIZED CLASS OF HARMONIC UNIVALENT FUNCTIONS INVOLVING A GAUSSIAN HYPERGEOMETRIC FUNCTION
The purpose of the present paper is to establish some new results giving the sharp bounds of the real parts of ratios of harmonic univalent functions to its sequences of partial sums by involving the Gaussian hypergeometric function. Relevant connections of the results presented here with various known results are briefly indicated. We also mention results which are associated with certain classical orthogonal polynomials (deduced from some of the main results) exhibiting the usefulness of the results presented in this paper.
Generalization of partial sums of certain analytic and univalent functions
Abstract Let ϕ ( z ) be a fixed analytic and univalent function of the form ϕ ( z ) = z + ∑ k = 2 ∞ c k z k and H ϕ ( c k , δ ) be the subclass consisting of analytic and univalent functions f ( z ) which satisfy the inequality ∑ k = 2 ∞ c k | a k | δ . In this paper, we study the ratio of a function of the form f ( z ) = z + ∑ k = 2 ∞ a k z k to its sequence of partial sums of the form f n ( z ) = z + ∑ k = 2 n a k z k where the coefficients of f ( z ) satisfy the above condition. Also, we determine sharp lower bounds for Re { f ( z ) / f n ( z ) } , Re { f n ( z ) / f ( z ) } , Re { f ′ ( z ) / f n ′ ( z ) } and Re { f n ′ ( z ) / f ′ ( z ) } .
Some Fox-Wright generalized hypergeometric functions and associated families of convolution operators
Here, in this lecture, we aim at presenting a systematic account of the basic properties and characteristics of several subclasses of analytic functions (with Montel’s normalization), which are based upon some convolution operators on Hilbert space involving the Fox-Wright generalization of the classical hypergeometric qFs function (with q numerator and s denominator parameters). The various results presented in this lecture include (for example) normed coefficient inequalities and estimates, distortion theorems, and the radii of convexity and starlikeness for each of the analytic function classes which are investigated here. We also briefly indicate the relevant connections of the some of the results considered here with those involving the Dziok-Srivastava operator.