AN INVESTIGATION ON GEOMETRIC PROPERTIES OF ANALYTIC FUNCTIONS WITH POSITIVE AND NEGATIVE COEFFICIENTS EXPRESSED BY HYPERGEOMETRIC FUNCTIONS

Abstract

This paper aims to investigate characterizations on parameters k(1), k(2), k(3), k(4), k(5), l(1), l(2), l(3), and l(4) to find relation between the class of H(k, l, m, n, o) hypergeometric functions defined by F-5(4) [(l1, l2, l3, l4) (k1, k2, k3, k4, k5,) : z] = Sigma(infinity)(n=2) (k(1))(n) (k(2))(n) (k(3))(n) (k(4))(n) (k(5))(n)/(l(1))(n) (l(2))(n) (l(3))(n) (l(4))(n) (1)(n) z(n). We need to find k, l, m and n that lead to the necessary and sufficient condition for the function zF([W]), G = z(2 - F ([W])) and H-1[W] =z(2) d/dz (ln(z) - h(z)) to be in S*(2(-r)), r is a positive integer in the open unit disc D = {z : vertical bar z vertical bar < 1, z is an element of C} with h(z) = Sigma(infinity)(n=0) (k)(n)(l)(n)(m)(n)(n)(n) (1 + k/2)(n)/(k/2)(n)(1 + k - l)(n) (1 + k - m)(n) (1 + k - n)(n)n(1)(n) z(n) and [W] = [(k/2, 1 + k -l, 1 + k - m, 1 + k - n) (k, 1 + k/2, l,) (m, n) :z].