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<p>This paper is devoted to continuous Opial-type inequalities involving the Riemann-Liouville fractional derivatives. The starting point was the inequality in which the integral of the product of the derivatives of one function is evaluated from above. This inequality is traditionally called the multiple Opial-type inequality. Here, we give a few generalizations of it. In the first type of generalization, the product of a finite number of factors is replaced by an expression involving infinitely many factors or even continuously many. Results involving <InlineEquation ID="IEq1"><EquationSource Format="MATHML"><math><msub><mi>L</mi><mi>p</mi></msub></math></EquationSource><EquationSource Format="TEX">$L_{p}$</EquationSource></InlineEquation> norm and sup norm in the estimations are obtained. Also, a reverse inequality is discussed. The second part of the paper concerns the inequality in which the integral of the product of successive derivatives of one function is bounded from above by the sum of integrals. Here, we make a similar generalization in which we transform the product of finitely many functions into its continuous form, which allows the result not only for finitely many functions but even for infinitely many functions.</p>

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Continuous form of the multiple Opial-type inequalities for the Riemann-Liouville fractional derivatives

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  • 发表日期：

    2026年7月16日

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  • Ilko Brnetić,
  • Ludmila Nikolova,
  • Sanja Varošanec

摘要

This paper is devoted to continuous Opial-type inequalities involving the Riemann-Liouville fractional derivatives. The starting point was the inequality in which the integral of the product of the derivatives of one function is evaluated from above. This inequality is traditionally called the multiple Opial-type inequality. Here, we give a few generalizations of it. In the first type of generalization, the product of a finite number of factors is replaced by an expression involving infinitely many factors or even continuously many. Results involving Lp$L_{p}$ norm and sup norm in the estimations are obtained. Also, a reverse inequality is discussed. The second part of the paper concerns the inequality in which the integral of the product of successive derivatives of one function is bounded from above by the sum of integrals. Here, we make a similar generalization in which we transform the product of finitely many functions into its continuous form, which allows the result not only for finitely many functions but even for infinitely many functions.