We consider the family of all analytic and univalent functions in the unit disk of the form f (z) = z+a 2 z 2 +a 3 z 3 +· · · . Our objective in this paper is to estimate the difference of the moduli of successive coefficients, that is |a n+1 | − |a n | , for f belonging to the family of γ-spirallike functions of order α. Our particular results include the case of starlike and convex functions of order α and other related class of functions.2010 Mathematics Subject Classification. 30D30, 30C45, 30C50 30C55.

where Γ 1 , Γ 2 , and Γ 3 are the first, second and third logarithmic coefficients of inverse functions belonging to the class S of normalized univalent functions. In this article, we establish sharp inequalitiesfor the logarithmic coefficients of inverse functions, considering starlike and convex functions, as well as functions with bounded turning of order 1/2, respectively.

We consider the family of all meromorphic functions f of the formanalytic and locally univalent in the puncture disk D 0 := {z ∈ C : 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 S f (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 + a 2 z 2 + a 3 z 3 + · · · analytic and locally univalent in the open unit disk D := {z ∈ C : |z| < 1}, and show that g is belonging to a family of functions convex in one direction if |S g (z)| is bounded above by a small positive constant depending on the second coefficient a 2 . In particular, we show that such functions g are also contained in the starlike and close-to-convex family.

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We consider a family of all analytic and univalent functions in the unit disk of the form f (z) = z + a 2 z 2 + a 3 z 3 + • • • . In this paper, we obtain the sharp bounds of the second Hankel determinant of Logarithmic coefficients for some subclasses of analytic functions. ∞ n=1 γ n (f )z n , z ∈ D.