CORRECTIONS TO ERRORS IN THE PAPER "HADAMARD-TYPE INEQUALITIES FOR <i>s</i>-CONVEX FUNCTIONS I" AND NEW INTEGRAL INEQUALITIES OF <i>s</i>-CONVEX FUNCTIONS IN THE SECOND SENSE

Jing-Yu Wang; Hong-Ping Yin; Bo-Yan Xi; Feng Qi

doi:10.11948/20240403

2025 Volume 15 Issue 5

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2025 > 15(5): 2652-2662

Next Article Previous Article

Jing-Yu Wang, Hong-Ping Yin, Bo-Yan Xi, Feng Qi. CORRECTIONS TO ERRORS IN THE PAPER 'HADAMARD-TYPE INEQUALITIES FOR s-CONVEX FUNCTIONS I' AND NEW INTEGRAL INEQUALITIES OF s-CONVEX FUNCTIONS IN THE SECOND SENSE[J]. Journal of Applied Analysis & Computation, 2025, 15(5): 2652-2662. doi: 10.11948/20240403

Citation:

Jing-Yu Wang, Hong-Ping Yin, Bo-Yan Xi, Feng Qi. CORRECTIONS TO ERRORS IN THE PAPER "HADAMARD-TYPE INEQUALITIES FOR s-CONVEX FUNCTIONS I" AND NEW INTEGRAL INEQUALITIES OF s-CONVEX FUNCTIONS IN THE SECOND SENSE[J]. Journal of Applied Analysis & Computation, 2025, 15(5): 2652-2662. doi: 10.11948/20240403

CORRECTIONS TO ERRORS IN THE PAPER "HADAMARD-TYPE INEQUALITIES FOR s-CONVEX FUNCTIONS I" AND NEW INTEGRAL INEQUALITIES OF s-CONVEX FUNCTIONS IN THE SECOND SENSE

1.
College of Mathematical Sciences, Inner Mongolia Minzu University, Tongliao 028043, China
2.
Institute of Nonlinear Analysis, Inner Mongolia Minzu University, Tongliao 028043, China
3.
School of Mathematics and Physics, Hulunbuir University, Hulunbuir 021008, Inner Mongolia, China
4.
17709 Sabal Court, Dallas, TX 75252-8024, USA
5.
School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454010, Henan, China

Author Bio: Email: jing-yu.wang@qq.com(J.-Y. Wang); Email: hongpingyin@qq.com(H.-P. Yin); Email: baoyintu78@qq.com(B.-Y. Xi)
Corresponding author: Email: qifeng618@gmail.com(F. Qi)

Abstract

Abstract

In the work, the authors correct some errors appeared in the paper “S. Hussain, M. I. Bhatti and M. Iqbal, Hadamard-type inequalities for s-convex functions I, Punjab Univ. J. Math. (Lahore), 41 (2009), 51–60” and establish some new integral inequalities of s-convex functions in the second sense.
- Hermite–Hadamard type inequality /
- correction /
- s-convex function /
- concave function
MSC: 26A51, 26D15, 26D20, 26E60, 41A55

FullText(HTML)

References

[1]	W. W. Breckner, Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen, Publ. Inst. Math. (Beograd) (N.S. ), 1978, 23(37), 13–20. (German) Google Scholar
[2]	B. Çelik, E. Set, A. O. Akdemir and M. E. Özdemir, Novel generalizations for Grüss type inequalities pertaining to the constant proportional fractional integrals, Appl. Comput. Math., 2023, 22(2), 275–291. Available online at https://doi.org/10.30546/1683-6154.22.2.2023.275. doi: 10.30546/1683-6154.22.2.2023.275 CrossRef Google Scholar
[3]	G. Gulshan, H. Budak, R. Hussain and K. Nonlaopon, Some new quantum Hermite-Hadamard type inequalities for s-convex functions, Symmetry, 2022, 14(5), Art. 870, 14 pp. Available online at https://doi.org/10.3390/sym14050870. doi: 10.3390/sym14050870 CrossRef Google Scholar
[4]	H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Math., 1994, 48(1), 100–111. Available online at http://dx.doi.org/10.1007/BF01837981. doi: 10.1007/BF01837981 CrossRef Google Scholar
[5]	S. Hussain, M. I. Bhatti and M. Iqbal, Hadamard-type inequalities for s-convex functions I, Punjab Univ. J. Math. (Lahore), 2009, 41, 51–60. Google Scholar
[6]	A. Karaoğlan, E. Set, A. O. Akdemir and M. E. Özdemir, Fractional integral inequalities for preinvex functions via Atangana–Baleanu integral operators, Miskolc Math. Notes, 2024, 25(1), 329–347. Available online at https://doi.org/10.18514/MMN.2024.4367. doi: 10.18514/MMN.2024.4367 CrossRef Google Scholar
[7]	M. A. Khan, Y.-M. Chu, T. U. Khan and J. Khan, Some new inequalities of Hermite-Hadamard type for s-convex functions with applications, Open Math., 2017, 15(1), 1414–1430. Available online at https://doi.org/10.1515/math-2017-0121. doi: 10.1515/math-2017-0121 CrossRef Google Scholar
[8]	U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula, Appl. Math. Comput., 2004, 147(1), 137–146. Available online at http://dx.doi.org/10.1016/S0096-3003(02)00657-4. doi: 10.1016/S0096-3003(02)00657-4 CrossRef Google Scholar
[9]	U. S. Kirmaci, M. Klaričić Bakula, M. E. Özdemir and J. Pečarić, Hadamard-type inequalities for s-convex functions, Appl. Math. Comput., 2007, 193(1), 26–35. Available online at http://dx.doi.org/10.1016/j.amc.2007.03.030. doi: 10.1016/j.amc.2007.03.030 CrossRef Google Scholar
[10]	M. Samraiz, M. Malik, S. Naheed and A. O. Akdemir, Error estimates of Hermite–Hadamard type inequalities with respect to a monotonically increasing function, Math. Methods Appl. Sci., 2023, 46(13), 14527–14546. Available online at https://doi.org/10.1002/mma.9334. doi: 10.1002/mma.9334 CrossRef Google Scholar
[11]	M. Z. Sarikaya, E. Set and M. E. Özdemir, On new inequalities of Simpson's type for s-convex functions, Comput. Math. Appl., 2010, 60(8), 2191–2199. Availble online at http://dx.doi.org/10.1016/j.camwa.2010.07.033. doi: 10.1016/j.camwa.2010.07.033 CrossRef Google Scholar
[12]	E. Set, A. O. Akdemir and A. Karaoğlan, New integral inequalities for synchronous functions via Atangana–Baleanu fractional integral operators, Chaos Solitons Fractals, 2024, 186, Paper No. 115193, 6 pp. Available online at https://doi.org/10.1016/j.chaos.2024.115193. doi: 10.1016/j.chaos.2024.115193 CrossRef Google Scholar
[13]	J.-Y. Wang, X.-M. Liu, H.-P. Yin and B.-N. Guo, Some new integral inequalities of Hermite–Hadamard type for functions whose third derivatives are (α, s, m)-convex, Miskolc Math. Notes, 2025, 26(1), in Press. Available online at https://doi.org/10.18514/MMN.2025.4756. Google Scholar
[14]	J.-Y. Wang, H.-P. Yin, W.-L. Sun and B.-N. Guo, Hermite–Hadamard's integral inequalities of (α, s)-GA-and (α, s, m)-GA-convex functions, Axioms, 2022, 11(11), Art. 616, 12 pp. Available online at https://doi.org/10.3390/axioms11110616. doi: 10.3390/axioms11110616 CrossRef Google Scholar
[15]	B.-Y. Xi, D.-D. Gao and F. Qi, Integral inequalities of Hermite–Hadamard type for (α, s)-convex and (α, s, m)-convex functions, Ital. J. Pure Appl. Math., 2020, 44, 499–510. Google Scholar
[16]	B.-Y. Xi, S.-H. Wang and F. Qi, Some inequalities for (h, m)-convex functions, J. Inequal. Appl., 2014, 2014, Art. 100, 12 pp. Available online at http://dx.doi.org/10.1186/1029-242X-2014-100. doi: 10.1186/1029-242X-2014-100 CrossRef Google Scholar
[17]	B.-Y. Xi, S.-H. Wang and F. Qi, Integral inequalities of Ostrowski type for two kinds of s-logarithmically convex functions, Georgian Math. J., 2024, 31(6), 1063–1071. Available online at http://dx.doi.org/10.1515/gmj-2024-2018. doi: 10.1515/gmj-2024-2018 CrossRef Google Scholar