On k–Airy convexity and applications to eigenvalue problems

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

Let Ai and Bi denote the Airy functions. For a fixed k > 0, we introduce the class Airy_k($\mathcal{I}$) of k–Airy convex functions on an interval $\mathcal{I} \subset[0, \infty)$. A function f : $\mathcal{I} \rightarrow \mathbb{R}$ is called k–Airy convex on $\mathcal{I}$ if, for every subinterval $[a, b] \subset \mathcal{I}$ and all a < t < b,

$f(t) \leq \frac{\mathrm{Ai}(k t) \operatorname{Bi}(k b)-\mathrm{Ai}(k b) \operatorname{Bi}(k t)}{\operatorname{Ai}(k a) \operatorname{Bi}(k b)-\operatorname{Ai}(k b) \operatorname{Bi}(k a)} f(a)+\frac{\operatorname{Ai}(k a) \operatorname{Bi}(k t)-\operatorname{Ai}(k t) \operatorname{Bi}(k a)}{\operatorname{Ai}(k a) \operatorname{Bi}(k b)-\operatorname{Ai}(k b) \operatorname{Bi}(k a)} f(b) .$

We establish fundamental properties of Airy_k($\mathcal{I}$). As an application, we derive lower bounds for eigenvalues in Airy-type boundary value problems.

Keywords

Airy functions / k–Airy convex functions / Airy differential equation / eigenvalue problems

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] J. M. Borwein, J. D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples, Cambridge: Cambridge University Press, 2010. https://doi.org/10.1017/CBO9781139087322

[2] C. Niculescu, L. E. Persson, Convex Functions and Their Applications, Springer: Berlin, Germany, 2006. https://doi.org/10.1007/978-3-031-71967-7

[3] A. W. Roberts, D. E. Varberg, Convex Functions, Academic Press, New York, NY, USA, 1973. Available from: https://books.google.com/books?id=moOaoAEACAAJ.

[4] S. Sra, R. Hosseini, Geometric optimisation on positive definite matrices for elliptically contoured distributions, in Advances in Neural Information Processing Systems, (2013), 2562-2570. Available from: https://proceedings.neurips.cc/paper/2013/hash/3948ead63a9f2944218de038d8934305-Abstract.html.

[5] K. J. Arrow, M. D. Intriligator, Handbook of Mathematical Economics, North-Holland, Amsterdam, 1981. Available from: https://library.wur.nl/WebQuery/titel/2178033.

[6] B. de Finetti, On convex stratifications, Ann. Mat. Pura Appl., 30(1949), 173-183. https://doi.org/10.1007/BF02415006

[7] S. S. Dragomir, Operator Schur convexity and some integral inequalities, Linear Multilinear Algebra, 69(2019), 2733-2748. https://doi.org/10.1080/03081087.2019.1694484

[8] H. Kadakal, Better approximations for quasi-convex functions, Stud. Univ. Babeş-Bolyai Math., 69(2024), 267-281. https://doi.org/10.24193/subbmath.2024.2.02

[9] Q. H. Xu, T. Jiang, T. S. Liu, The refinement of Fekete and Szegö problems for close-to-convex functions and close-to-quasi-convex mappings, J. Math. Anal. Appl., 527(2023), 127428. https://doi.org/10.1016/j.jmaa.2023.127428

[10] W. Orlicz, A note on modular spaces I, Bull. Acad. Polon. Soi., Ser. Math. Astronom Phys., 9(1961), 157-162.

[11] W. W. Breckner, Continuity statements for a class of generalized convex functions in topological linear spaces, Publ. Inst. Math., 23(1978), 13-20. Available from: http://eudml.org/doc/257486.

[12] S. Aslan, Ü. Demir, E. Karaduman, A. Akdemir, Some novel fractional Milne-type inequalities for twice differentiable s-convex functions in the second sense, Filomat, 39(2025), 3425-3435. https://doi.org/10.2298/FIL2510425A

[13] P. Kórus, An extension of the Hermite-Hadamard inequality for convex and s-convex functions, Aequ. Math., 93(2019), 527-534. https://doi.org/10.1007/s00010-019-00642-z

[14] Y. Zhao, H. Sang, W. Xiong, Z. Cui, Hermite-Hadamard-type inequalities involving ψ-Riemann-Liouville fractional integrals via s-convex functions, J. Inequal. Appl, 2020(2020), 128. https://doi.org/10.1186/s13660-020-02389-7

[15] G. H. Toader, Some generalisations of the convexity, in Proceedings of Colloquium on Approximation and Optimization, Romania, (1984), 329-338.

[16] S. S. Dragomir, On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math., 33(2002), 45-55. https://doi.org/10.5556/j.tkjm.33.2002.304

[17] S. S. Dragomir, G. Toader, Some inequalities for m-convex functions, Studia Univ. Babes-Bolyai Math., 38(1993), 21-28.

[18] S. Varošanec, On h-convexity, J. Math. Anal. Appl., 326(2007), 303-311. https://doi.org/10.1016/j.jmaa.2006.02.086

[19] B. Benaissa, N. Azzouz, H. Budak, Weighted fractional inequalities for new conditions on h-convex functions, Bound. Value Probl., 2024(2024), 76. https://doi.org/10.1186/s13661-024-01889-5

[20] B. Benaissa, N. Azzouz, H. Budak, Hermite-Hadamard type inequalities for new conditions on h-convex functions via ψ-Hilfer integral operators, Anal. Math. Phys., 14(2024), 35. https://doi.org/10.1007/s13324-024-00893-3

[21] L. Zhang, Y. Peng, T. Du, On multiplicative Hermite-Hadamard-and Newton-type inequalities for multiplicatively (P, m)-convex functions, J. Math. Anal. Appl., 534(2024), 128117. https://doi.org/10.1016/j.jmaa.2024.128117

[22] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, U.S. Government Printing Office, Washington, 1964. Available from: https://archive.org/details/handbookofmathem1964abra.

[23] O. Vallée, M. Soares, Airy Functions and Applications to Physics, World Scientific Publishing Company, 2010. https://doi.org/10.1142/p709

[24] M. H. Protter, H. F. Weinberger, Maximum Principles in Differential Equations, New York: Springer, 1984. https://doi.org/10.1007/978-1-4612-5282-5

[25] A. Lyapunov, General problem of the stability of motion, Ann. Fac. Sci. Toulouse, 9(1907), 204-474. Available from: http://eudml.org/doc/72801.

[26] S. S. Dragomir, C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, Australia, 2000. Available from: https://rgmia.org/papers/monographs/Master.pdf.

[27] H. Budak, F. Hezenci, T. Tunç, H. Kara, On new versions of Hermite-Hadamar-type inequalities based on tempered fractional integrals, Filomat, 38(2024), 2361-2379. https://doi.org/10.2298/FIL2407361B

[28] S. S. Dragomir, B. T. Torebek, Some Hermite-Hadamard type inequalities in the class of hyperbolic p-convex functions, RACSAM, 113(2019), 3413-3423. https://doi.org/10.1007/s13398-019-00708-2

[29] H. Kara, M. A. Ali, H. Budak, Hermite-Hadamard-type inequalities for interval-valued coordinated convex functions involving generalized fractional integrals, Math. Methods Appl. Sci., 44(2021), 104-123. https://doi.org/10.1002/mma.6712