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Article

Second-Order Differential Inequality Convexity

1
Department of Mathematical, Physical and Chemical Sciences, University of Zagreb, Croatian Academy of Sciences and Arts, 10000, Zagreb, Croatia
2
Institute for Sustainable Industries and Liveable Cities, Victoria University, Melbourne 3001, Australia
*
Author to whom correspondence should be addressed.
Axioms 2026, 15(5), 330; https://doi.org/10.3390/axioms15050330
Submission received: 5 March 2026 / Revised: 12 April 2026 / Accepted: 27 April 2026 / Published: 1 May 2026

Abstract

Some equivalent statements and basic properties for the generalized convexity assumption p(x)f(x)+q(x)f(x)+f(x)0 are proved. Then based on these conclusions, various Jensen type inequalities under the generalized convexity are established. The idea is to transform such p(x),q(x)-convex functions to some simpler p(x),0-convex or 0,q(x)-convex functions, or even convex functions. Ky Fan and Wang-Wang type inequalities are also generalized as applications.
Keywords: differential inequality; convex function; Jensen inequality differential inequality; convex function; Jensen inequality

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MDPI and ACS Style

Pečarić, J.; Miao, J. Second-Order Differential Inequality Convexity. Axioms 2026, 15, 330. https://doi.org/10.3390/axioms15050330

AMA Style

Pečarić J, Miao J. Second-Order Differential Inequality Convexity. Axioms. 2026; 15(5):330. https://doi.org/10.3390/axioms15050330

Chicago/Turabian Style

Pečarić, Josip, and Jinyan Miao. 2026. "Second-Order Differential Inequality Convexity" Axioms 15, no. 5: 330. https://doi.org/10.3390/axioms15050330

APA Style

Pečarić, J., & Miao, J. (2026). Second-Order Differential Inequality Convexity. Axioms, 15(5), 330. https://doi.org/10.3390/axioms15050330

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