Local Fractional Hilbert-Type Inequalities with a Non-Homogeneous Kernel

Predrag Vuković

doi:10.52737/18291163-2025.17.7-1-13

Authors

Predrag Vuković University of Zagreb

DOI:

https://doi.org/10.52737/18291163-2025.17.7-1-13

Keywords:

Hilbert Inequality, Conjugate Parameters, Homogeneous Function, Local Fractional Calculus

Abstract

The main objective of this paper is to study some new local fractional Hilbert-type inequalities with a general kernel. We apply our main results to non-homogeneous kernels. In addition, we obtain the best possible constants.

References

D. Baleanu, M. Krnić and P. Vuković, A class of fractal Hilbert-typeinequalities obtained via Cantor-type spherical coordinates. Math. Meth. Appl. Sci., 44 (2021), no. 7, pp. 6195-6208.

Y. Basci and D. Baleanu, Hardy-type inequalities within fractional derivatives without singular kernel. J. Inequal. Appl., 2018 (2018), Article number 304.

G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities. Second edition. Cambridge University Press, Cambridge, 1967.

G. Jumarie, Fractional Euler's integral of first and second kinds. Application to fractional Hermite's polynomials and to probability density of fractional order. J. Appl. Math. & Informatics, 28 (2010), no. 1-2, pp. 257-273.

M. Krnić, J. Pečarić, I. Perić and P. Vuković, Recent advances in Hilbert-type inequalities, Element, Zagreb, 2012.

M. Krnić and P. Vuković, Multidimensional Hilbert-type inequalities obtained via local fractional calculus. Acta Appl. Math., 169 (2020), pp. 667-680.

Q. Liu, A Hilbert-type fractional integral inequality with the kernel of Mittag-Leffler function and its applications. Math. Inequal. Appl., 21 (2018), no. 3, pp. 729-737.

T. Liu, R.A. Rahim and B. Yang, A new Hilbert-type integral inequality with the general nonhomogeneous kernel and applications. J. Math. Inequal., 17 (2023), no. 4, pp. 1581-1596.

M.Th. Rassias and B. Yang, On an equivalent property of a peverse Hilbert-type integral inequality related to the extended Hurwitz-zeta function. J. Math. Inequal., 13 (2019), no. 2, pp. 315-334.

M.Th. Rassias and B. Yang, On a Hilbert-type integral inequality in the whole plane related to the extended Riemann zeta function. Complex Anal. Oper. Theory, 13 (2019), pp. 1765-1782.

M.Th. Rassias, B. Yang and G.C. Meletiou, A more accurate half-discrete Hilbert-type inequality in the whole plane and the reverses. Ann. Funct. Anal., 12 (2021), Article number 50.

M.Th. Rassias, B. Yang and A. Raigorodskii, On a more accurate reverse Hilbert-type inequality in the whole plane. J. Math. Inequal., 14 (2020), no. 4, pp. 1359-1374.

M.Th. Rassias, B. Yang and A. Raigorodskii, A Hilbert-type integral inequality in the whole plane related to the arc tangent function. Symmetry, 13 (2021), no. 2, Article number 351.

M.Th. Rassias, B. Yang and A. Raigorodskii, Equivalent conditions of a multiple Hilbert-type integral inequality with the nonhohogeneous kernel. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 116 (2022), Article number 107.

M.Th. Rassias, B. Yang and A. Raigorodskii, An equivalent form related to a Hilbert-type integral inequality. Axioms, 12 (2023), no.7, Article number 677.

P. Vuković, Some local fractional Hilbert-type inequalities. Fractal Fract., 7 (2023), no. 2, Article number 205.

X.J. Yang, Local Fractional Functional Analysis and its Applications, Asian Academic publisher Limited, Hong Kong, 2011.

X.J. Yang, Advanced Local Fractional Calculus and its Applications, Word Science Publisher, New York, 2012.

B. Yang, D. Andrica, D. Bagdasar and M.Th. Rassias, An equivalent property of a Hilbert-type integral inequality and its applications. Appl. Anal. Discrete Math., 16 (2022), no. 2, pp. 548-563.

B. Yang, D. Andrica, O. Bagdasar and M.Th. Rassias, On a Hilbert-type integral inequality in the whole plane with the equivalent forms. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 117 (2023), Article number 57.

B. Yang and M.Th. Rassias, On Hilbert-Type and Hardy-Type Integral Inequalities and Applications, Springer Cham, 2019.