We prove a generalized version of Hermite–Hadamard–Mercer type inequalities using the beta function. Moreover, we prove some new trapezoidal type inequalities involving the beta function for differentiable convex functions. The main advantage of these inequalities is that these can be converted into similar classical integral inequalities, Riemann–Liouville fractional integral inequalities and $k$-Riemann–Liouville fractional integral inequalities. Finally, we give applications to special means of real numbers for newly established inequalities.

We consider the family of all analytic and univalent functions in the unit disk of the form $f(z)=z+a_2{z}^2+a_3{z}^3+\cdots$. We obtain the sharp bounds of the second-Hankel determinant of logarithmic coefficients for some subclasses of analytic functions.

We aim to prove generalized estimations for the $q$-Hermite–Hadamard inequality for convex functions using a parameter. By choosing this particular parameter, we show that our results reduce the previously obtained $q$-Hermite–Hadamard inequalities. We also present some new refinements of these $q$-Hermite–Hadamard inequalities. Furthermore, we establish a new lemma and, by using this lemma, we obtain some new quantum inequalities that generalize quantum midpoint and quantum trapezoid inequalities for convex functions.

Two $g$-frames ${\{{\mathrm{\Lambda }}_i\}}_{i\in \mathcal{ℐ}}$ and ${\{{\mathrm{\Gamma }}_i\}}_{i\in \mathcal{ℐ}}$ for the Hilbert space $\mathcal{𝒰}$ are called woven if for each subset $\sigma$ of $\mathcal{ℐ}$ the weaving ${\{{\mathrm{\Lambda }}_i\}}_{i\in \sigma }\cup {\{{\mathrm{\Gamma }}_i\}}_{i\in {\sigma }^c}$ is a $g$-frame for $\mathcal{𝒰}$. The aim of this paper is to consider the reordered families of a $g$-frame ${\{{\mathrm{\Lambda }}_i\}}_{i\in \mathcal{ℐ}}$ with various woven problems. First, we state some useful results for exact $g$-frames and excess of $g$-frames. Then for $\sigma \subset \mathcal{ℐ}$ we consider the families of weavings ${\{{\mathrm{\Lambda }}_i\}}_{i\in \sigma }\cup {\{{\mathrm{\Lambda }}_{\pi (i)}\}}_{i\in {\sigma }^c}$ where $\pi$ is a permutation function on $\mathcal{ℐ}$ and obtain some new conclusions. Lastly, we give relations of reordered weavings and operators, especially $g$-frame operators.

A class of examples is constructed to show that for strictly stationary Markov chains that are reversible, the simultaneous mixing rates for the $\rho$-mixing and strong mixing ($\alpha$-mixing) conditions can be fairly arbitrary, within certain unavoidable tight restrictions. The examples constructed here have the added property that the mixing rate for the absolute regularity ($\beta$-mixing) condition is within a constant factor of that for strong mixing.

Applying the triplicate form of the extended Gould–Hsu inverse series relations to Dougall’s summation theorem for the well-poised ${}_7{F}_6$-series, we establish, from the dual series, several interesting Ramanujan-like infinite series expressions for ${\pi }^2$ and ${\pi }^{\pm 1}$ with convergence rate $-\frac{1}{27}$.

The study of positive-definite matrices has focused on Hermitian matrices, that is, square matrices with complex (or real) entries that are equal to their own conjugate transposes. In the classical setting, positive-definite matrices enjoy a multitude of equivalent definitions and properties. We investigate when a square, symmetric matrix with entries coming from a finite field can be called “positive-definite” and discuss which of the classical equivalences and implications carry over.

With the help of a newly defined contraction operator, we establish a fixed point theorem and study the solvability of fractional hybrid differential equations in a Banach space. We provide examples to support our findings.

We study the distance between permutations in three different settings which are related to DNA and quantum entanglements. We construct graphs where the vertices correspond to permutations of enhanced permutations and edges are defined by adjacent permutations to define distances. Numerous bounds and a recursion formula are given for these distances. These distances are then related to distances in the Braid group.

The construction of a suitable Lyapunov function is still a difficult task. This paper mainly analyzes the practical uniform exponential stability of linear time-varying singular systems, which are transferable into standard canonical form. Our method is based on the explicit solution form of the system via integral inequalities of the type of Gamidov under some restrictions of the perturbation term. An example is analyzed to verify the effectiveness of the proposed approach.

We consider the weighted composition-differentiation operator $D_{{\psi }_n,{\phi }_n,n}$ on the Hardy and weighted Bergman spaces. We describe the spectrum and the spectral radius of the operator $D_{{\psi }_n,{\phi }_n,n}$. Also the lower estimate and the upper estimate on the norm of the weighted composition-differentiation operator on the Hardy space $H^2$ are obtained. Furthermore, we determine the norm of some composition-differentiation operators $D_{\phi ,n}$ on the Hardy space $H^2$.

Fix a base $b\ge 2$ and an exponent $e\ge 2$. An $e$-power $b$-happy number is a positive integer that reaches $1$ under iteration of the function mapping a positive integer to the sum of the $e$-th powers of its base $b$ digits. In this note, we answer the question of how large the gaps between $e$-power $b$-happy numbers can be.

A natural problem is to determine, for each value of the integer $g\ge 2$, the largest order of a group that acts on a Riemann surface of genus $g$. Let $N(g)$ (respectively $M(g)$) be the largest order of a group of automorphisms of a Riemann surface of genus $g\ge 2$ preserving the orientation (respectively possibly reversing the orientation) of the surface.

Let $g=1+3p$ for a large prime $p$. It has been established that if $p$ is congruent to $1(\text{mod}6)$, then $N(g)=M(g)=24(g-1)$. Suppose $p$ is congruent to $5(\text{mod}6)$. We prove that if $p$ is also congruent modulo 25 to 1, 6, 11 or 16, then $N(g)=8(g+11)$ and $M(g)=16(g+11)$; otherwise $N(g)=8(g+1)$ and $M(g)=16(g+1)$.

We deal with the time-fractional Navier–Stokes equations with damping in a bounded domain $\mathrm{\Omega }$ in ${ℝ}^3$. First, we establish the existence of weak solutions by Galerkin approximation for $\beta \ge 1$. We also show the uniqueness of weak solutions for $\beta \ge 4$. Further, we prove the regularity of the solution for $\beta \ge 3$ and .

We show every isometric action of a countable discrete group on a compact space is quasidiagonal in a strong sense. This shows that reduced crossed products by such actions are quasidiagonal or MF whenever the reduced group $C^{\ast }$-algebra of the acting group is quasidiagonal or MF. We use this to show new examples of group actions whose crossed products are MF.

Given a sense-preserving harmonic function defined in the open unit disk with its analytic part restricted to the class of starlike functions, several techniques are developed to construct stable close-to-convex harmonic mappings. Each technique is demonstrated through illustrations. Moreover, the radii of full convexity are computed for sense-preserving harmonic functions with the analytic part belonging to certain subclasses of univalent functions. The obtained bounds are sharp.

We consider the $\rho$-weighted $p$-norm of functions $f:{ℝ}_{+}\to ℝ$ with finite $q$-norm $\parallel f^{(n)}\psi {\parallel }_q,1\le q\le +\infty$. A sharp Wirtinger type inequality

$$\parallel f\rho {\parallel }_p\le C_{n,p,q}\parallel f^{(n)}\psi {\parallel }_q\text{ for all}1\le p,q\le +\infty$$

is established for functions $f$ such that $f^{(j)}(x_i)=0$ for all $0\le j\le {\alpha }_i-1$, $i=1,\dots ,r$, $n={\sum }_{i=1}^r{\alpha }_i$, where $\psi$, $\rho$ and $\omega =\rho ∕\psi$ are nonincreasing on ${ℝ}_{+}$, and ${\omega }^{1∕\alpha }$ is integrable for $\alpha =n-1∕q+1∕p$. Using Hermite interpolation, we express $C_{n,p,q}$ in terms of the norm of a certain integral type operator. Then we calculate $C_{n,1,1}$ and $C_{n,\infty ,\infty }$ in two specific cases.

Eldred, Kirk and Veeramani introduced the notion of proximal normal structure to prove the best proximity point theorems for relatively nonexpansive mappings and showed that uniformly convex Banach spaces have proximal normal structure. In this paper, a characterization for the weak proximal normal structure is given. Using this characterization, it is proved that every weakly compact convex pair in a Banach space $X$ has proximal normal structure whenever $X$ satisfies: $X$ is ${\epsilon }_0$-inquadrate in every direction for some ${\epsilon }_0\in (0,1)$ or $X$ has the modulus of $k$-UC , for $k\in \mathbb{N}$ or $X$ has the modulus of $k$-dimensional $U$-convexity , for $k\in \mathbb{N}$ or $X$ has the coefficient of noncompact convexity ${\epsilon }_1(X)<1$. Moreover, we generalize the notion ${\epsilon }_0$-inquadrate in every direction to ${\epsilon }_0$-inquadrate with respect to every $k$-dimensional subspace and showed that $X$ has the weak proximal normal structure and the weak normal structure if ${\epsilon }_0\in (0,1)$. In the case of ${\epsilon }_0\in [1,2)$, the Banach space $X$ has weak proximal normal structure with an additional assumption $X$ has the WORTH property.

The synchronization problem for fractional-order Jafari–Sprott chaotic systems is investigated. Two primary synchronization control laws are proposed to achieve the chaotic synchronization of the system without/with uncertainties. For the simple fractional-order Jafari–Sprott chaotic system without considering uncertainties, active control is proposed to achieve chaotic synchronization. In addition, a robust synchronization control is realized for the corresponding chaotic system with uncertainties, by adopting a fractional-order sliding mode control strategy. The controller parameters could be adjusted based on the stability theory of the fractional-order system and the fractional-order sliding mode theory, respectively. As illustrated by numerical examples, the proposed active control law and slide mode control strategy could work well and accurately.

We introduce von Neumann–Schatten $K$-$p$-frames and von Neumann–Schatten $K^{\ast }$-atomic systems, where $K$ is a bounded operator on a separable Banach space, and we collect some relationships between these two concepts. Also, we characterize von Neumann–Schatten $K$-$p$-frames in terms of a range inclusion property and show that they are stable under different kinds of perturbations and the action of some bounded operators. Moreover, we introduce and characterize the notions of $K$-duals and $K^{\ast }$-duals and obtain some of their properties.

The main purpose of this paper is to study upper-bound error estimates (also known as error bounds) for a class of generalized double phase obstacle problems via regularized gap functions. More precisely, we introduce some regularized gap functions for the double phase obstacle problem in forms introduced by Yamashita and Fukushima, and apply these regularized gap functions to provide the upper-bound error estimates for the double phase obstacle problem.

By applying Ricceri’s variational principle, we demonstrate the existence of solutions for a double weighted Robin problem in weighted variable exponent Sobolev spaces under some appropriate conditions.

Let $\lfloor x\rfloor$ be the largest integer not exceeding $x$. For , let ${\pi }_{𝜃}(x)$ denote the number of integers $n$ with $1\le n\le x^{𝜃}$ such that $\lfloor x∕n\rfloor$ is prime. Recently, Ma, Chen and Wu obtained the interesting asymptotic formula

$${\pi }_{𝜃}(x)=\frac{x^{𝜃}}{(1-𝜃)\text{log}x}+O(x^{𝜃}{(\text{log}x)}^{-2}),$$

provided that . They further conjectured that this asymptotic formula can be extended to all . In this paper, we give an improvement of their result by showing that is admissible.