Open research questions in Mathematical Inequalities and Applications

What the literature leaves open

• Future research could explore further applications of post-quantum integrals. The study suggests that further research could investigate the properties of n-polynomial pre-invex functions.

  On Novel Extension of n-Polynomial Pre-invex Functions and Related Post-Quantum Integral Inequalities with Application · 2026 · DOI

• The lack of a unified approach to Ostrowski-type inequalities on arbitrary time scales. The need for explicit estimates for the deviation of a function value from the mean delta integral on arbitrary time scales. The gap in the existing literature regarding the extension of the Ostrowski inequality to time scales.

  Two-Sided Ostrowski-Type Inequalities on Time Scales Under Integrable Derivative Bounds · 2026 · DOI

• The classical theory of Hilbert-type inequalities lacks richer analytical frameworks. The paper identifies a gap in the existing literature on Hilbert-type integral inequalities.

  Study of Two New Hilbert-Type Integral Inequalities with Arctangent-Maximum-Geometric Mean Kernel Functions · 2026 · DOI

• The need for a new integral identity based on Riemann-Liouville fractional operators. The lack of applications of fractional corrected dual Euler-Simpson inequalities to inequalities involving different types of means.

  Some Fractional Corrected Dual Euler-Simpson Type Inequalities for Differentiable Strongly Preinvex Functions · 2026 · DOI

• Existing inequalities for the beta function and its ratios are not sharp. There is a need for new methods to prove inequalities for the beta function and its ratios. The paper identifies a gap in existing inequalities for the beta function and its ratios.

  Some inequalities for the beta function and its ratios · 2026 · DOI

• Further research could explore the applications of the established inequalities. Further research could investigate the extensions of the results to other contexts.

  Affine Logarithmic HLS and Beckner-Type Logarithmic Sobolev Inequalities · 2026 · DOI

• The paper suggests several directions for future research, including the extension of fractional Hermite-Hadamard type inequalities to weighted settings. The study proposes the exploration of other generalized classes of convex functions beyond the m-convex class. The results can be adapted to the discrete setting, providing analogous inequalities in discrete calculus.

  Generalization and refinement of fractional Hermite-Hadamard type inequalities for m-convex functions · 2026 · DOI

• The extension of the main results in case of other fractional integrals remains open. The paper identifies a gap in the generalization of some existing results.

  Hermite-Hadamard type inequalities via \((h,m)\)-convexity · 2026 · DOI

• Future research can focus on extending the results to other types of convex functions. Future research can focus on applying the results to other areas of mathematics. Future research can focus on deriving new inequalities using the results presented in the paper.

  Jensen-Type Inequalities for Divided Differences via Generalized Convex Functions · 2026 · DOI

• The definition of l-gap convex functions is more complex than that of convex functions. The proof of the Jensen type inequality for l-gap convex functions is challenging.

  Jensen Type Inequality for L-Gap Convex Functions · 2026 · DOI

• There is a need for a generalization of convex functions. The existing definitions of convex functions are not sufficient to model certain real-world problems.

  Jensen Type Inequality for L-Gap Convex Functions · 2026 · DOI

• The paper suggests potential extensions to Hilbert space operators and matrix means inequalities. Further research can focus on applying the results to various fields, such as linear algebra and operator theory.

  Mean inequalities for norms and singular values · 2026 · DOI

• There is a need to study the properties of close-to-convex functions defined using the beta negative binomial distribution. The Fekete-Szegö inequality has not been derived for this class of functions.

  Classes of close-to-convex Functions Defined using Beta Negative Binomial Distribution · 2026 · DOI

• Future research can focus on applying the results to various fields such as complex analysis and functional analysis. Future research can explore the properties of bounded holomorphic mappings in the polydisk.

  Multidimensional analogues of the Refined Bohr type inequalities · 2026 · DOI

• The gap in the existing literature is the lack of multidimensional analogues of the Refined Bohr type inequalities. The paper identifies the need for new sharp inequalities for bounded holomorphic mappings in the polydisk.

  Multidimensional analogues of the Refined Bohr type inequalities · 2026 · DOI

• The paper does not provide a complete characterization of the boundary of the cone of almost-entropic vectors. The paper leaves open the problem of closing the remaining gap between the upper bound O(√ε) and the lower bound Ω(ε^2/3).

  Structural properties of entropic vectors and stability of the Ingleton inequality · 2026 · DOI

• Future research should focus on closing the remaining gap between the upper bound O(√ε) and the lower bound Ω(ε^2/3). Future research should explore the implications of the paper's results for the study of conditional independence structures.

  Structural properties of entropic vectors and stability of the Ingleton inequality · 2026 · DOI

• The authors note that existing inequalities can be improved using refined techniques.

  Mean inequalities for norms and singular values · 2026 · DOI

• The general version of the change of variable theorem is not typically presented. The theorem is not typically included in introductory texts on analysis.

  The general change of variable formula for the Riemann integral · 2026 · DOI

• The paper identifies a gap in the literature regarding the analysis of non-convex functions. The study notes that non-convex and non-differentiable objective functions display generalized convex-like behavior.

  On Novel Extension of n-Polynomial Pre-invex Functions and Related Post-Quantum Integral Inequalities with Application · 2026 · DOI

• Investigate the case 0 < y ≤ 1 in more detail, - Conduct experimental validation of the results, - Provide a comprehensive comparison with existing results…

  Some inequalities for the beta function and its ratios · 2026 · DOI

• The authors identify a gap in the existing literature on affine inequalities. The authors aim to fill this gap by establishing an affine logarithmic HLS inequality and an affine version of Beckner's logarithmic Sobolev inequality.

  Affine Logarithmic HLS and Beckner-Type Logarithmic Sobolev Inequalities · 2026 · DOI

• The paper identifies a gap in the literature on Bernstein inequalities on the unit ball. The paper aims to fill this gap by establishing new Bernstein inequalities.

  On Bernstein Inequalities on the Unit Ball · 2026 · DOI

• The paper identifies a gap in the existing literature regarding Jensen-Type Inequalities for Divided Differences via Generalized Convex Functions. The paper aims to fill this gap by deriving an estimate for Jensen’s inequality in the context of divided differences using (m+4)-convex functions.

  Jensen-Type Inequalities for Divided Differences via Generalized Convex Functions · 2026 · DOI

• Further study of the connection between elliptic and parabolic monotonicity formulas. Application of the results to other areas of mathematics and physics.

  Deficit functions and the log Sobolev inequality · 2026 · DOI

Most-cited papers in Mathematical Inequalities and Applications

• The Three Sigma Rule · The American Statistician · 1994 · 1,855 citations
• Measures of Inequality · American Sociological Review · 1978 · 807 citations
• Differentiation Under the Integral Sign · American Mathematical Monthly · 1973 · 558 citations
• The Price of Inequality · New Perspectives Quarterly · 2013 · 535 citations
• On an Extension of the Gini Inequality Index · International Economic Review · 1983 · 429 citations
• The Prehistory of the Hardy Inequality · American Mathematical Monthly · 2006 · 400 citations
• The Logarithmic Mean · American Mathematical Monthly · 1972 · 373 citations
• Generalized gini inequality indices · Mathematical Social Sciences · 1981 · 339 citations
• Bonnesen-Style Isoperimetric Inequalities · American Mathematical Monthly · 1979 · 293 citations
• On Limiting the Domain of Inequality · The Journal of Law and Economics · 1970 · 272 citations

Most recent work

• Radau-type inequalities from multiplicative Katugampola fractional integrals theoretical point of view · Journal of Computational and Applied Mathematics · 2026
• Characterization of polyconvex isotropic functions · Calculus of Variations and Partial Differential Equations · 2026
• On the exact region between Chatterjee’s rank correlation and Spearman’s footrule · Journal of Computational and Applied Mathematics · 2026
• Robust low-rank matrix completion via joint schatten capped p-norm and capped ℓp-norm · Applied Mathematical Modelling · 2026
• Structural properties of entropic vectors and stability of the Ingleton inequality · Information Geometry · 2026
• Triangle inequalities for the operator symmetric modulus · Proceedings of the American Mathematical Society · 2026
• Fractional integral inequalities for exponentially preinvex functions via conformable fractional integrals · Journal of Computational and Applied Mathematics · 2026
• ON MULTIPLICATIVE FRACTIONAL OPERATORS OF HADAMARD AND KATUGAMPOLA TYPES IN G-CALCULUS AND RELATED HERMITE–HADAMARD INEQUALITIES · Fractals · 2026
• Characterizations and linearity problem of the weighted spectral geometric mean · Linear Algebra and its Applications · 2026
• Log-majorizations between quasi-geometric type means for matrices · Linear Algebra and its Applications · 2026

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