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    STUDIA MATHEMATICA - Issue no. 3 / 2018  
         
  Article:   MEROMORPHIC FUNCTIONS WITH SMALL SCHWARZIAN DERIVATIVE.

Authors:  VIBHUTI ARORA, SWADESH KUMAR SAHOO.
 
       
         
  Abstract:  
 We consider the family of all meromorphic functions f of the form

analytic and locally univalent in the puncture disk D0 := {zC : 0 < |z| < 1}. Our first objective in this paper is to find a sufficient condition for f to be meromorphically convex of order α, 0 ≤ α < 1, in terms of the fact that the absolute value of the well-known Schwarzian derivative Sf(z) of f is bounded above by a smallest positive root of a non-linear equation. Secondly, we consider a family of functions g of the form g(z) = z + a2z2 + a3z3 + ··· analytic and locally univalent in the open unit disk D := {zC : |z| < 1}, and show that g is belonging to a family of functions convex in one direction if |Sg(z)| is bounded above by a small positive constant depending on the second coefficient a2. In particular, we show that such functions g are also contained in the starlike and close-to-convex family.

Mathematics Subject Classification (2010): 30D30, 30C45, 30C55, 34M05. 
Keywords: Meromorphic functions, convex functions, meromorphically convex functions, close-to-convex functions, starlike functions, Schwarzian derivative.

 
         
     
         
         
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