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We begin the study of Hankel matrices whose entries are logarithmic coefficients of univalent functions and give sharp bounds for the second Hankel determinant of logarithmic coefficients of convex and starlike functions.

Let $F:(0, \infty) \times [0, \infty) \rightarrow \mathbb{R}$ be a function of Carathéodory type. We establish the inequality $$ \int_{\mathbb{R}^{N}} F( | x |, u(x) ) dx \leq \int_{\mathbb{R}^{N} } F( | x |, u^{\ast}(x)) dx. $$ where $u^{\ast}$ denotes the Schwarz symmetrization of $u$, under hypotheses on $F$ that seem quite natural when this inequality is used to obtain existence results in the context of elliptic partial differential equations. We also treat the case where $\mathbb{R}^N$ is replaced by a set of finite measure. The identity $$ \int_{\mathbb{R}^{N}} G(u(x)) dx = \int_{\mathbb{R}^{N}} G(u^{\ast}(x)) dx $$ is also discussed under the assumption that $G : [0,\infty) \rightarrow \mathbb{R}$ is a Borel function.

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SECOND HANKEL DETERMINANT OF LOGARITHMIC COEFFICIENTS OF CONVEX AND STARLIKE FUNCTIONS

Symmetrization Inequalities for Composition Operators of Carathéodory type

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SECOND HANKEL DETERMINANT OF LOGARITHMIC COEFFICIENTS OF CONVEX AND STARLIKE FUNCTIONS

Symmetrization Inequalities for Composition Operators of Carathéodory type