On α-convex functions related to shell-like functions connected with Fibonacci numbers

• Published in: Applied mathematics and computation 2011-10, Vol.218 (3), p.996-1002
• Main Authors: Dziok, Jacek, Raina, Ravinder Krishna, Sokół, Janusz
• Format: Article
• Language: English
• Subjects: Convex functions; Fibonacci numbers; Starlike functions; Subordination; Trisectrix of Maclaurin; Univalent functions; α-Convex functions; α-Convex functions of order β
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2011.01.059

This paper presents a new class SLM α of functions f( z) analytic and normalized in the open unit disc U = { z : | z | < 1 } (which is related to a shell-like curve and associated with Fibonacci numbers) satisfying the condition that α 1 + zf ″ ( z ) f ′ ( z ) + ( 1 - α ) zf ′ ( z ) f ( z ) ∈ p ˜ ( U ) ( z ∈ U ) , where α is a real number and p ˜ ( z ) = τ z + τ 2 z 2 1 - τ z - τ 2 z 2 ( τ = ( 1 - 5 ) / 2 ; z ∈ U ) . The class SLM α being closely related to the classes of starlike and convex functions, we apply some basic techniques to investigate certain interesting properties (given below) for this class of functions. Some important observations of the main results are also mentioned.