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    AN INVESTIGATION ON GEOMETRIC PROPERTIES OF ANALYTIC FUNCTIONS WITH POSITIVE AND NEGATIVE COEFFICIENTS EXPRESSED BY HYPERGEOMETRIC FUNCTIONS
    (Honam Mathematical Soc, 2022) Akyar, Alaattin; Mert, Oya; Yıldız, İsmet
    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].
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    Convex and Starlike Functions Defined on the Subclass of the Class of the Univalent Functions S with Order 2(- r)
    (Univ Maragheh, 2022) Yıldız, İsmet; Mert, Oya; Akyar, Alaattin
    In this paper, some conditions have been improved so that the function g(z) is defined as g(z) = 1+ Sigma(infinity)(k >= 2) alpha n+k(zn+k), which is analytic in unit disk U, can be in more specific subclasses of the S class, which is the most fundamental type of univalent function. It is analyzed some characteristics of starlike and convex functions of order 2(-r).