TY - JOUR
TI - Certain results of starlike and convex functions in some conditions
AU - Yildiz Ismet
AU - Sahin Hasan
JN - Thermal Science
PY - 2022
VL - 26
IS - 102
SP - 719
EP - 725
PT - Article
AB - The theory of geometric functions was first introduced by Bernard Riemann in  1851. In 1916, with the concept of normalized function revealed by  Bieberbach, univalent function concept has found application area. Assume f(z)=z+Σ∞, n≥(anzn)  converges for all complex numbers z with |z|<1 and f(z)is one-to-one on the set of  such z. Convex and starlike functions f(z) and g(z) are discussed with the  help of subordination. The f(z) and g(z) are analytic in unit disc and f(0)=f'(0)=1, and  g(0)=0, g'(0)-1=0. A single valued function f(z) is said to be univalent  (or schlict or one-to-one) in domain D⊂C never gets the same value twice; that  is, if f(z1)-f(z2)≠0 for all z1 and z2 with z1 ≠ z2. Let A be the class of analytic  functions in the unit disk U={z:|z|<1} that are normalized with f(0)=F'(0)=1. In this paper we give  the some necessary conditions for f(z) ∈ S* [a, a2] and 0≤a2≤a≤1 f'(z)(2r-1)[1-f'(z)]+zf"(z / 2r[f'(z)]2. This condition means that convexity and  starlikeness of the function f of order 2-r.