Difference Between the Initial Logarithmic Coefficients for Certain Univalent Functions

For a normalized univalent function f(z) = z + ∑ _n=1^∞ a_n z^n defined in the unit disc 𝔻 , the coefficients γ _n determined by the expansion log (f(z)/z ) = 2 ∑ _n=1^∞γ _n z^n are called the logarithmic coefficients of f . In this paper, sharp bounds for the quantity |γ _3 |- |γ _2 | is established, where γ _2 and γ _3 denote the second and third logarithmic coefficients, respectively, for functions belonging to the following subclasses: starlike functions of order α ; close-to-convex functions satisfying Re( 1 + z f”(z)/f'(z) ) > α , functions satisfying Re (1 + z f”(z)/f'(z) ) < 1 + α /2 ; and functions of bounded turning with Ref'(z) > α . For the class of bounded turning functions, the estimate of | γ _2| -| γ _1| is also discussed.