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Register a new userSharp inequalities for logarithmic coefficients and their applications
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I. M. Milin proposed, in his 1971 paper, a system of inequalities for the logarithmic coefficients of normalized univalent functions on the unit disk of the complex plane. This is known as the Lebedev-Milin conjecture and implies the Robertson conjecture which in turn implies the Bieberbach conjecture. In 1984, Louis de Branges settled the long-standing Bieberbach conjecture by showing the Lebedev-Milin conjecture. Recently, O.~Roth proved an interesting sharp inequality for the logarithmic coefficients based on the proof by de Branges. In this paper, following Roths ideas, we will show more general sharp inequalities with convex sequences as weight functions and then establish several consequences of them. We also consider the inequality with the help of de Branges system of linear ODE for non-convex sequences where the proof is partly assisted by computer. Also, we apply some of those inequalities to improve previously known results.
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We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding soltions with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with lower bounds characterized by the geometry of the manifold. The geometric invariant appearing in the sharp lower bound is shown to be nonnegative. We also characterize the expanders when such invariant is zero. In the proof various useful volume growth estimates are also established for gradient shrinking and expanding solitons. In particular, we prove that the {it asymptotic volume ratio} of any gradient shrinking soliton with nonnegative Ricci curvature must be zero.
We consider interpolation inequalities for imbeddings of the $l^2$-sequence spaces over $d$-dimensional lattices into the $l^infty_0$ spaces written as interpolation inequality between the $l^2$-norm of a sequence and its difference. A general method is developed for finding sharp constants, extremal elements and correction terms in this type of inequalities. Applications to Carlsons inequalities and spectral theory of discrete operators are given.
Let $es$ be the class of analytic and univalent functions in the unit disk $|z|<1$, that have a series of the form $f(z)=z+ sum_{n=2}^{infty}a_nz^n$. Let $F$ be the inverse of the function $fines$ with the series expansion %in a disk of radius at least $1/4$ $F(w)=f^{-1}(w)=w+ sum_{n=2}^{infty}A_nw^n$ for $|w|<1/4$. The logarithmic inverse coefficients $Gamma_n$ of $F$ are defined by the formula $logleft(F(w)/wright),=,2sum_{n=1}^{infty}Gamma_n(F)w^n$. % In this paper, we determine the logarithmic inverse coefficients bound of $F$ for the class In this paper, we first determine the sharp bound for the absolute value of $Gamma_n(F)$ when $f$ belongs to $es$ and for all $n geq 1$. This result motivates us to carry forward similar problems for some of its important geometric subclasses. In some cases, we have managed to solve this question completely but in some other cases it is difficult to handle for $ngeq 4$. For example, in the case of convex functions $f$, we show that the logarithmic inverse coefficients $Gamma_n(F)$ of $F$ satisfy the inequality [ |Gamma_n(F)|,le , frac{1}{2n} mbox{ for } ngeq 1,2,3 ] and the estimates are sharp for the function $l(z)=z/(1-z)$. Although this cannot be true for $nge 10$, it is not clear whether this inequality could still be true for $4leq nleq 9$.
Let $es$ be the family of analytic and univalent functions $f$ in the unit disk $D$ with the normalization $f(0)=f(0)-1=0$, and let $gamma_n(f)=gamma_n$ denote the logarithmic coefficients of $fin {es}$. In this paper, we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the families $F(c)$ and $G(delta)$ of functions $fin {es}$ defined by $$ {rm Re} left ( 1+frac{zf(z)}{f(z)}right )>1-frac{c}{2}, mbox{ and } , {rm Re} left ( 1+frac{zf(z)}{f(z)}right )<1+frac{delta}{2},quad zin D $$ for some $cin(0,3]$ and $deltain (0,1]$, respectively. We obtain the sharp upper bound for $|gamma_n|$ when $n=1,2,3$ and $f$ belongs to the classes $F(c)$ and $G(delta)$, respectively. The paper concludes with the following two conjectures: begin{itemize} item If $finF (-1/2)$, then $ displaystyle |gamma_n|le frac{1}{n}left(1-frac{1}{2^{n+1}}right)$ for $nge 1$, and $$ sum_{n=1}^{infty}|gamma_{n}|^{2} leq frac{pi^2}{6}+frac{1}{4} ~{rm Li,}_{2}left(frac{1}{4}right) -{rm Li,}_{2}left(frac{1}{2}right), $$ where ${rm Li}_2(x)$ denotes the dilogarithm function. item If $fin G(delta)$, then $ displaystyle |gamma_n|,leq ,frac{delta}{2n(n+1)}$ for $nge 1$. end{itemize}
The notion of different kind of algebraic Casorati curvatures are introduced. Some results expressing basic Casorati inequalities for algebraic Casorati curvatures are presented. Equality cases are also discussed. As a simple application, basic Casorati inequalities for different $delta $-Casorati curvatures for Riemannian submanifolds are presented. Further applying these results, Casorati inequalities for Riemannian submanifolds of real space forms are obtained. Finally, some problems are presented for further studies.
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