Initial logarithmic coefficients for functions starlike with respect to symmetric points

Abstract

In this paper, we obtain the bounds of the initial logarithmic coefficients for functions in the classes $${\mathcal {S}}_S^*$$ S S ∗ and $${\mathcal {K}}_S$$ K S of functions which are starlike with respect to symmetric points and convex with respect to symmetric points, respectively. In our research, we use a different approach than the usual one in which the coeffcients of f are expressed by the corresponding coeffcients of functions with positive real part. In what follows, we express the coeffcients of f in $${\mathcal {S}}_S^*$$ S S ∗ and $${\mathcal {K}}_S$$ K S by the corresponding coeffcients of Schwarz functions. In the proofs, we apply some inequalities for these functions obtained by Prokhorov and Szynal, by Carlson and by Efraimidis. This approach offers a additional benefit. In many cases, it is easily possible to predict the exact result and to select extremal functions. It is the case for $${\mathcal {S}}_S^*$$ S S ∗ and $${\mathcal {K}}_S$$ K S .

Sharp Bounds for the Second Hankel Determinant of Logarithmic Coefficients for Strongly Starlike and Strongly Convex Functions

“…Hence by (12) we have a 2 = a 4 = 0 and a 3 = α. This shows that equality is attained for the function given in (10).…”

“…and are unknown for k ≥ 3. Recently, logarithmic coefficients have been studied by various authors and upper bounds of logarithmic coefficients of functions in some important subclasses of S have been obtained (e.g., [5][6][7][8][9][10]). For a summary of some of the significant results concerning the logarithmic coefficients for univalent functions, we refer to [11].…”

Second Hankel determinant of the logarithmic coefficients for a subclass of univalent functions

“…Recently, logarithmic coefficients were studied by many authors and the upper bounds of the logarithmic coefficients of functions in some important subclasses of the univalent function class S were found (see, for example, [1,2,4,6,16,22,25]). For a summary of some of the significant results concerning the bounds of the logarithmic coefficients for univalent functions, we refer to [24].…”

On the Second Hankel determinant of Logarithmic Coefficients for certain univalent functions

“…The problem of computing the bound of the logarithmic coefficients is also considered in [6,18,21] for several subclasses of close to convex functions. In 2021, Zaprawa [22] obtained the sharp bounds of the initial logarithmic coefficients |γ n | for functions in the classes S * S and K S .…”