Ramanujan's formula for the logarithmic derivative of the gamma function

On the Sobolev space of functions with derivative of logarithmic order

Two notions of "having a derivative of logarithmic order" have been studied. They come from the study of regularity of flows and renormalized solutions for the transport and continuity equation associated to weakly differentiable drifts.

Certain identities on derivatives of radial homogeneous and logarithmic functions

Let $k$ be a natural number and $s$ be real. In the 1-dimensional case, the $k$-th order derivatives of the functions $\lvert x\rvert^s$ and $\log \lvert x\rvert$ are multiples of $\lvert x\rvert^{s-k}$ and $\lvert x\rvert^{-k}$, respectively. In the present paper, we generalize this fact to higher dimensions by introducing a suitable norm of the derivatives, and give the exact values of the multiples.

Difference analogue of the Lemma on the Logarithmic Derivative with applications to difference equations

The Lemma on the Logarithmic Derivative of a meromorphic function has many applications in the study of meromorphic functions and ordinary differential equations. In this paper, a difference analogue of the Logarithmic Derivative Lemma is presented and then applied to prove a number of results on meromorphic solutions of complex difference equations. These results include a difference analogue of the Clunie Lemma, as well as other results on the value distribution of solutions.

A Newman type bound for $L_p[-1,1]$-means of the logarithmic derivative of polynomials having all zeros on the unit circle

Let $g_n$, $n=1,2,\dots$, be the logarithmic derivative of a complex polynomial having all zeros on the unit circle, i.e., a function of the form $g_n(z)=(z-z_{1})^{-1}+\dots+(z-z_{n})^{-1}$, $|z_1|=\dots=|z_n|=1$. For any $p>0$, we establish the bound \[\int_{-1}^1 |g_n(x)|^p\, dx>C_p\, n^{p-1},\] sharp in the order of the quantity $n$, where $C_p>0$ is a constant, depending only on $p$. The particular case $p=1$ of this inequality can be considered as a stronger variant of the well-known estimate $\iint_{|z|<1} |g_n(z)|\,dxdy>c>0$ for the area integral of $g_n$, obtained by D.J. Newman (1972…

关于对数导数的实临界点数目与Hawaii猜想

On the number of real critical points of logarithmic derivatives and the Hawaii conjecture

本文研究了实整函数对数导数的实零点数目关系，通过建立$Q$与$Q_1$实零点数目的联系，成功证明了Hawaii猜想。

Simpson Type Inequalities Via m- and (alpha,m)- logarithmically Convex functions

In this paper, we obtain some Simpson type inequalities for functions whose derivatives in absolute value are m- and (alpha,m)- logarithmically convex functions.

Inner Functions with Derivatives in the Weak Hardy Space

It is proved that exponential Blaschke products are the inner functions whose derivative is in the weak Hardy space. Exponential Blaschke products are described in terms of their logarithmic means and also in terms of the behavior of the derivatives of functions in the corresponding model space.

第1部分。与 Whittaker 函数 $\mathrm{M}_{κ,μ}(x)$ 关于参数微分相关的无穷级数和对数积分

Part 1. Infinite series and logarithmic integrals associated to differentiation with respect to parameters of the Whittaker $\mathrm{M}_{κ,μ}\left( x\right) $ function

研究 Whittaker 函数 $\mathrm{M}_{κ,μ}(x)$ 的参数一阶导数，给出其无穷级数与有限、无穷对数积分表示，并推导相关闭式约化公式。

修正贝塞尔函数乘积及其对数导数的尖锐双侧界

Sharp two-sided bounds for the product of modified Bessel functions and for its logarithmic derivative

本文为修正贝塞尔函数乘积及其对数导数之和建立了带最优常数的尖锐双侧不等式，证明基于辅助函数的二阶微分不等式与强极大值原理，并应用于乘积单调性和比值改进。

第2部分。与 Whittaker 函数 $\mathrm{W}_{κ,μ}\left(x\right)$ 关于参数微分相关的无穷级数和对数积分

Part 2. Infinite series and logarithmic integrals associated to differentiation with respect to parameters of the Whittaker $\mathrm{W}_{κ,μ}\left( x\right) $ function

研究 Whittaker 函数关于参数的一阶导数，给出其无穷级数与对数积分表示及特定参数下的闭合形式，并推导积分 Whittaker 函数的表示和约化公式。

Guo-Zhang-Qi猜想与digamma幂族的正Laplace核

The Guo-Zhang-Qi conjecture and positive Laplace kernels for a digamma power family

本文通过Laplace核正性分析，独立证明了Guo-Zhang-Qi关于digamma幂族对数完全单调性的猜想，并刻画了最优参数阈值。

On the logarithm of the derivative operator

We study the properties of the logarithm of the derivative operator and show that its action on a constant is not zero, but yields the sum of the logarithmic function and the Euler-Mascheroni constant. We discuss more general aspects concerning the logarithm of an operator for the study of the properties of the Bessel functions.

Fractional Poincare and logarithmic Sobolev inequalities for measure spaces

We prove generalizations of the Poincare and logarithmic Sobolev inequalities corresponding to the case of fractional derivatives in measure spaces with only a minimal amount of geometric structure. The class of such spaces includes (but is not limited to) spaces of homogeneous type with doubling measures. Several examples and applications are given, including Poincare inequalities for graph Laplacians, Fractional Poincare inequalities of Mouhot, Russ, and Sire [16], and implications for recent work of the author and R. M. Strain on the Boltzmann collision operator [10, 11, 9].

Exact bounds on the inverse Mills ratio and its derivatives

The inverse Mills ratio is $R:=φ/Ψ$, where $φ$ and $Ψ$ are, respectively, the probability density function and the tail function of the standard normal distribution. Exact bounds on $R(z)$ for complex $z$ with $\Re z\ge0$ are obtained, which then yield logarithmically exact bounds on high-order derivatives of $R$. The main idea of the proof is a non-asymptotic version of the so-called stationary-phase method.