关于对数导数的实临界点数目与Hawaii猜想

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

We prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in Ramanujan's notebooks. The formula has a number of very interesting consequences which we derive, including an elegant hyperbolic summation, Ramanujan's formula for the Riemann zeta function evaluated at the odd positive integers, and new formulae for Euler's constant, gamma.

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…

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.

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.

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.

Bounds for the logarithm of the Euler gamma function and its derivatives

We consider differences between $\log Γ(x)$ and truncations of certain classical asymptotic expansions in inverse powers of $x-λ$ whose coefficients are expressed in terms of Bernoulli polynomials $B_n(λ)$, and we obtain conditions under which these differences are strictly completely monotonic. In the symmetric cases $λ=0$ and $λ=1/2$, we recover results of Sonin, Nörlund and Alzer. Also we show how to derive these asymptotic expansions using the functional equation of the logarithmic derivative of the Euler gamma function, the representation of $1/x$ as a difference $F(x+1)-F(x)$, and a back…