# Weaker Conditions for the q-Steffensen Inequality and Some Related Generalizations

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**Theorem**

**1.**

**Remark**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

## 2. Weaker Conditions for q-Steffensen’s Inequality

**Lemma**

**1.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

## 3. Generalizations of the q-Steffensen Inequality

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Remark**

**2.**

**Theorem**

**11.**

**Proof.**

**Theorem**

**12.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**13.**

**Proof.**

**Theorem**

**14.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Theorem**

**15.**

**Theorem**

**16.**

**Remark**

**3.**

**Remark**

**4.**

## 4. Concluding Remarks

**Theorem**

**17.**

## Funding

## Conflicts of Interest

## References

- Kac, V.; Cheung, P. Quantum Calculus; Springer: New York, NY, USA, 2002. [Google Scholar]
- Thomae, J. Beiträge zur Theorie der durch die Heinesche Reihe: Darstellbaren Functionen. J. Reine Angew. Math.
**1869**, 1869, 258–281. [Google Scholar] - Jackson, F.H. On q-definite integrals. Quart. J. Pure Appl. Math.
**1910**, 41, 193–203. [Google Scholar] - Gauchman, H. Integral inequalities in q-calculus. Comput. Math. Appl.
**2004**, 47, 281–300. [Google Scholar] [CrossRef][Green Version] - Steffensen, J.F. On certain inequalities between mean values and their application to actuarial problems. Scand. Actuar. J.
**1918**, 1918, 82–97. [Google Scholar] [CrossRef] - Pečarić, J.; Smoljak Kalamir, K.; Varošanec, S. Steffensen’s and Related Inequalities (A Comprehensive Survey and Recent Advances); Monograhps in inequalities 7; Element: Zagreb, Croatia, 2014. [Google Scholar]
- Marinković, S.; Rajković, P.; Stanković, M. The inequalities for some types of q-integrals. Comput. Math. Appl.
**2008**, 56, 2490–2498. [Google Scholar] [CrossRef][Green Version] - Rajković, P.; Stanković, M.; Marinković, S.; Kirane, M. On q-Steffensen inequality. Electron. J. Differ. Equ.
**2018**, 2018, 112. [Google Scholar] - Iddrisu, M.M. q-Steffensen’s inequality for convex functions. Int. J. Math. Appl.
**2018**, 6, 157–162. [Google Scholar] - Milovanović, G.; Pečarić, J. The Steffensen inequality for convex function of order n. Univ. Beogr. Publ. Elektrotehn. Fak. Ser. Mat. Fiz.
**1979**, 634–677, 97–100. [Google Scholar] - Pečarić, J. Notes on some general inequalities. Publ. Inst. Math.
**1982**, 32, 131–135. [Google Scholar] - Mercer, P.R. Extensions of Steffensen’s inequality. J. Math. Anal. Appl.
**2000**, 246, 325–329. [Google Scholar] [CrossRef][Green Version] - Pečarić, J.; Perušić, A.; Smoljak, K. Mercer and Wu-Srivastava generalisations of Steffensen’s inequality. Appl. Math. Comput.
**2013**, 219, 10548–10558. [Google Scholar] [CrossRef] - Wu, S.-H.; Srivastava, H.M. Some improvements and generalizations of Steffensen’s integral inequality. Appl. Math. Comput.
**2007**, 192, 422–428. [Google Scholar] [CrossRef] - Pečarić, J.; Smoljak Kalamir, K. On some bounds for the parameter λ in Steffensen’s inequality. Kyungpook Math. J.
**2015**, 55, 969–981. [Google Scholar] [CrossRef][Green Version]

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Smoljak Kalamir, K. Weaker Conditions for the *q*-Steffensen Inequality and Some Related Generalizations. *Mathematics* **2020**, *8*, 1462.
https://doi.org/10.3390/math8091462

**AMA Style**

Smoljak Kalamir K. Weaker Conditions for the *q*-Steffensen Inequality and Some Related Generalizations. *Mathematics*. 2020; 8(9):1462.
https://doi.org/10.3390/math8091462

**Chicago/Turabian Style**

Smoljak Kalamir, Ksenija. 2020. "Weaker Conditions for the *q*-Steffensen Inequality and Some Related Generalizations" *Mathematics* 8, no. 9: 1462.
https://doi.org/10.3390/math8091462