The Pečarić Journal of Mathematical Inequalities (PJMI)

In this work, two enhanced versions of Wirtinger’s inequality are developed. These improvements arise when considering a weighted sum of multiple Wirtinger’s inequalities. Depending on the context, one of the proposed refinements may be applicable than the other. Finally, a simple application of such refinements is presented.

The Hardy-Hilbert integral inequality has inspired a vast body of research over the past few decades, resulting in the creation of numerous new forms and generalizations of integral inequalities. In this article, we build on this line of research by introducing a new class of Hardy-Hilbert-type integral inequalities incorporating an adjustable function. This additional flexibility enables our results to bridge the gap naturally between classical cases and a variety of new ones. We provide several distinct examples to illustrate the applicability and sharpness of the derived inequalities. Additionally, we present a supplementary result that extends the main theorem, supported by concrete examples that demonstrate its validity and scope.

We present a new sharp Ostrowski-type inequality in the L₂ norm for functions with absolutely continuous second derivative and third derivative in L₂. The inequality depends on two parameters α, γ ∈ [0, 1] and generalizes the sharp inequality of Liu [1]. Special choices of parameters yield known sharp inequalities for midpoint, trapezoid, Simpson, corrected Simpson, and averaged midpoint-trapezoid rules. A complete sharpness proof is given, including explicit verification of the extremal function’s regularity. Applications to composite numerical integration are provided with explicit error bounds, and a numerical example illustrates the theoretical estimates.

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.