The Pečarić Journal of Mathematical Inequalities (PJMI)

This article introduces and analyzes a new class of integral inequalities relating the integrals of two functions over different intervals. Using classical tools such as the Hermite-Hadamard, Steffensen and Young integral inequalities, we derive several refined bounds under monotonicity and convexity assumptions.

This article makes a contribution to the ongoing development of the Steffensen integral inequality by presenting two new results. The first result generalizes the classical Steffensen integral inequality by introducing an additional function that combines key aspects of the Steffensen and Chebyshev integral inequalities. The second result presents a concave integral inequality derived using integration techniques. Numerical examples are provided to demonstrate the validity and application of the results.

This article presents a two-dimensional extension of divisibility networks, constructed on generalized integer lattice, and developed to explore their applications in inequality structures. Edges join nodes that share either the same multiple index \(n\) or the same divisor index \(k\), which form a rook–divisibility network that unites arithmetic structure and graph topology within a deterministic grid. The resulting finite graph \(G_N=\{(k,n)\in\mathbb{N}^2:1\le n\le N,\,k\mid n\}\) admits exact analysis of its main invariants. Closed forms are derived for the local degree \(\deg(k,n)\) and clustering coefficient \(C(k,n)\); they reveal how small \(k\) columns act as hubs and highly composite rows yield strong local cohesion. A constructive proof via projection maps establishes global connectivity for all \(N\), and asymptotic evaluation shows that the average degree grows as \(\langle k\rangle_N\!\sim\!(\pi^2/6)\,N/\log N\), much faster than in the one-dimensional divisor network. The results provide a heavy–tailed degree distribution governed by a logarithmic factor, while empirical simulations and log-binned spectra confirm close agreement between measured and analytic clustering across degree ranges. Further visual analyses illustrate the emergence of hubs, stretching similarity, and stable scaling of local clustering. In addition, the rook-divisibility framework is shown to generate new forms of discrete and fractional inequalities. By interpreting row- and column-averaging operations as convex and fractional mean processes, the model yields Hermite-Hadamard-Mercer-type bounds and degree-clustering inequalities.

This article presents a generalization of the Hardy–Littlewood–Pólya majorization theorem by employing a weighted Montgomery identity derived from Taylor’s formula. We establish new identities and inequalities for n-convex functions, provide Čebyšev-type bounds for the remainders, and derive associated Ostrowski and Grüss-type inequalities. Our results significantly extend the classical theory of majorization and provide a comprehensive framework for analyzing n-convex functions in the context of weighted integral inequalities.

This work is dedicated to some generalized upper bounds obtained for the Jensen gap by using different generalized convex functions including strongly convex functions, s-convex functions, η-convex functions, strongly η-convex functions, m-convex functions, and (α, m)-convex functions. The results are then extended to the integral form of Jensen’s inequality. The main results enable us to establish such bounds for Hölder and Hermite-Hadamard inequalities as well. Finally, estimates for the Csiszár divergence are presented as direct applications of the main outcomes.

This paper deals with the finite element approximation of the elliptic impulse control quasi-variational inequality (QVI), when the impulse control cost goes to zero. By means of the concepts of subsolutions for QVIs and a Lipschitz dependence property with respect to the impulse cost, an \(L^{\infty}\) error estimate is derived for both the impulse control QVI and the correponding asymptotic problem.

In this paper, we give extensions of Jensen-Mercer inequality for functions whose derivatives in the absolute values are uniformly convex considering the class of \(k-\)fractional integral operators.

This paper investigates the application of the Hardy-Littlewood-Sobolev inequality to analyze the global existence of solutions for a wave equation incorporating distributed delay effects and Hartree-type nonlinearities under suitable analytical conditions.

We study a coercive quasi-variational inequality (QVI) system and propose a generalized Schwarz method using finite element approximations. The discrete solution is iteratively constructed through monotone upper and lower sequences, and its convergence is rigorously established in the \(\mathfrak{L}^\infty\) norm. This framework ensures stability, geometric convergence, and efficient computation on overlapping subdomains.

Let \(\left( H;\left\langle \cdot ,\cdot \right\rangle \right)\) be a complex Hilbert space. Denote by \(\mathcal{B}\left( H\right)\) the Banach \(C^{\ast }\) -algebra of bounded linear operators on \(H\). For \(A\in \mathcal{B}\left( H\right)\) we define the modulus of \(A\) by \(\left\vert A\right\vert :=\left( A^{\ast }A\right) ^{1/2}.\) We say that the continuous function \(B:\left[ a,b \right] \rightarrow \mathcal{B}\left( H\right)\) is square modulus convex (concave) on \(\left[ a,b\right]\) if \[\begin{equation*} \left\vert B\left( \left( 1-t\right) u+tv\right) \right\vert ^{2}\leq \left( \geq \right) \left( 1-t\right) \left\vert B\left( u\right) \right\vert ^{2}+t\left\vert B\left( v\right) \right\vert ^{2}, \end{equation*}\] in the operator order of \(\mathcal{B}\left( H\right) ,\) for all \(u,\) \(v\in \left[ a,b\right]\) and \(t\in \left[ 0,1\right] .\) In this paper, we show among others that, if \(B:\left[ m,M\right] \subset \mathbb{R\rightarrow } \mathcal{B}\left( H\right)\) is square modulus convex on \(\left[ m,M\right]\) and \(f:\Omega \rightarrow \left[ m,M\right]\) so that \(f,\) \(\left\vert B\circ f\right\vert ^{2},\) \(Re\left( \left( B\circ f\right) ^{\ast }\left( B^{\prime }\circ f\right) \right) ,\) \(fRe\left( \left( B\circ f\right) ^{\ast }\left( B^{\prime }\circ f\right) \right) \in L_{w}\left( \Omega ,\mu ,\mathcal{B}\left( H\right) \right) ,\) where \(w\geq 0\) \(\mu\) -a.e. on \(\Omega\) with \(\int_{\Omega }wd\mu =1,\) then \[0 \leq \int_{\Omega }w\left( s\right) \left\vert B\circ f\right\vert ^{2}d\mu \left( s\right) -\left\vert B\left( \int_{\Omega }wfd\mu \right) \right\vert ^{2}\] \[\qquad\qquad\qquad~\qquad\qquad \leq \frac{1}{2}\left( M-m\right) \left\Vert \left( Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) – Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) \right) \right\Vert .\] The discrete versions are also provided.