CERTAIN MULTIVALENT HARMONIC FUNCTION DEFINED BY GENERALIZED HYPERGEOMETRIC FUNCTIONS
VIMLESH KUMAR GUPTA *
Department of Mathematics & Astronomy, University of Lucknow, Lucknow, 226007 U.P., India.
*Author to whom correspondence should be addressed.
Abstract
In this paper, some multivalent harmonic functions F = H +
∈ H (m) which are defined in terms of generalized hypergeometric functions are considered. Making use of coefficient inequalities for F ∈ H (m) satisfying some well known class conditions, we obtain the hypergeometric inequalities, which are sufficient for F to be in the classes S*H(m, α),KH(m, α) and QH(m, α) and are necessary for the harmonic function F1 ∈ TH(m) ⊂ H (m). Further, some necessary and sufficient hypergeometric inequalities are given under some parametric restriction such that F ∈ TH(m). Results based on some integral operators are also given. A concluding remark based on special cases is mentioned.
Keywords: Multivalent harmonic starlike (convex) functions, generalized hypergeometric functions, gauss hypergeometric functions