# Generalized Preinvex Functions and Their Applications

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**([1])**

- 1.
- a b-vex function on K with respect to (w.r.t. in short) b if$$h\left(\right)open="("\; close=")">t{u}_{1}+(1-t){u}_{2}$$
- 2.
- and a b-linear function on K w.r.t. b if$$h\left(\right)open="("\; close=")">t{u}_{1}+(1-t){u}_{2}$$

**Definition**

**2**

**([2])**

**Definition**

**3**

**([3])**

**Definition**

**4**

**([6])**

**Definition**

**5.**

## 3. Sub-b-s-Preinvex Function and Their Properties

**Definition**

**6.**

**Remark**

**7.**

- 1.
- 2.
- When $\eta ({u}_{1},{u}_{2})={u}_{1}-{u}_{2}$ and $b(u,u,t)\le 0$ in Equation (1), then the sub-b-s-preinvex function becomes a convex function.

**Theorem**

**8.**

**Corollary**

**9.**

**Proposition**

**10.**

**Theorem**

**11.**

- 1.
- ${h}_{2}\left(\right)open="("\; close=")">\beta {u}_{1}$;
- 2.
- ${h}_{2}\left(\right)open="("\; close=")">{u}_{1}+{u}_{2}$.

**Proof.**

**Definition**

**12.**

**Theorem**

**13.**

**Proof.**

**Proposition**

**14.**

**Proof.**

**Proposition**

**15.**

**Theorem**

**16.**

- 1.
- $d{h}_{{u}_{2}}\eta ({u}_{1},{u}_{2})\le {t}^{s-1}\left(\right)open="("\; close=")">h\left({u}_{1}\right)+h\left({u}_{2}\right)$;
- 2.
- $d{h}_{{u}_{2}}\eta ({u}_{1},{u}_{2})\le {t}^{s-1}\left(\right)open="("\; close=")">h\left({u}_{1}\right)-h\left({u}_{2}\right)$.

**Proof.**

- By using the hypothesis, we can write$$h\left(\right)open="("\; close=")">{u}_{2}+t\eta ({u}_{1},{u}_{2})$$Additionally,$$h\left(\right)open="("\; close=")">{u}_{2}+t\eta ({u}_{1},{u}_{2})$$Furthermore,$$\begin{array}{c}\hfill h\left(\right)open="("\; close=")">{u}_{2}+t\eta ({u}_{1},{u}_{2})\\ \le & {t}^{s}h\left({u}_{1}\right)+{(1-t)}^{s}h\left({u}_{2}\right)+b({u}_{1},{u}_{2},t)\hfill \end{array}$$Then$$h\left({u}_{2}\right)+td{h}_{{u}_{2}}\eta ({u}_{1},{u}_{2})+O\left(t\right)\le {t}^{s}h\left({u}_{1}\right)+(1+{t}^{s})h\left({u}_{2}\right)+b({u}_{1},{u}_{2},t)$$
- Similarly,$$\begin{array}{c}h\left({u}_{2}\right)+td{h}_{{u}_{2}}\eta ({u}_{1},{u}_{2})+O\left(t\right)\hfill \\ \phantom{\rule{36.135pt}{0ex}}\le {t}^{s}h\left({u}_{1}\right)+(1+{t}^{s})h\left({u}_{2}\right)+b({u}_{1},{u}_{2},t)\hfill \\ \phantom{\rule{36.135pt}{0ex}}={t}^{s}h\left({u}_{1}\right)+(1+{t}^{s})h\left({u}_{2}\right)-{t}^{s}h\left({u}_{2}\right)+{t}^{s}h\left({u}_{2}\right)+b({u}_{1},{u}_{2},t)\hfill \\ \phantom{\rule{36.135pt}{0ex}}={t}^{s}\left(\right)open="("\; close=")">h\left({u}_{1}\right)-h\left({u}_{2}\right)+b({u}_{1},{u}_{2},t)+\left(\right)open="("\; close=")">{(1-t)}^{s}+{t}^{s}\hfill & h\left({u}_{2}\right).\end{array}$$However, we know that ${(1-t)}^{s}+{\delta}^{s},\forall t\in [0,1]$, and $s\in \left(\right)open="("\; close>\left(\right)open\; close="]">0,1$ and since h is non-negative function; hence,$$h\left({u}_{2}\right)+\delta {h}_{{u}_{2}}\eta ({u}_{1},{u}_{2})+O\left(t\right)\le {t}^{s}\left(\right)open="("\; close=")">h\left({u}_{1}\right)-h\left({u}_{2}\right)$$Then, by dividing the last inequality by t and taking ${lim}_{t\u27f6{0}_{+}}$, we obtain the second part of the theorem.

**Theorem**

**17.**

**Proof.**

**Corollary**

**18.**

- 1.
- h is a non-negative function, then$$d\left(\right)open="("\; close=")">{h}_{{u}_{2}}-{h}_{{u}_{1}}$$
- 2.
- h is a negative function, then$$d\left(\right)open="("\; close=")">{h}_{{u}_{2}}-{h}_{{u}_{1}}$$

**Proof.**

- Since h is a non-negative function and by using Theorem 16,$$d{h}_{{u}_{2}}\eta ({u}_{1},{u}_{2})\le {t}^{s-1}\left(\right)open="("\; close=")">h\left({u}_{1}\right)-h\left({u}_{2}\right)$$Additionally,$$d{h}_{{u}_{1}}\eta ({u}_{1},{u}_{2})\le {t}^{s-1}\left(\right)open="("\; close=")">h\left({u}_{2}\right)-h\left({u}_{1}\right)$$Thus,$$d\left(\right)open="("\; close=")">{h}_{{u}_{2}}-{h}_{{u}_{1}}$$
- Since h is a negative function, and according to Theorem 17, the second result can be obtained directly.

## 4. Hermite–Hadamard-Type Integral Inequalities for Differentiable Sub-B-S-Preinvex Functions

**Theorem 19.**

**Theorem**

**20.**

**Lemma**

**21**

**([27])**

**Theorem**

**22.**

**Proof.**

**Corollary**

**23.**

**Theorem**

**24.**

**Proof.**

**Corollary**

**25.**

## 5. Application

**Theorem**

**26.**

**Proof.**

**Example**

**27.**

**Corollary**

**28.**

**Proof.**

**Theorem**

**29**

**(Karush-Kuhn-Tucker**

**Sufficient**

**Conditions)**

**Proof.**

- The arithmetic mean:$$A=A({u}_{1},{u}_{2})=\frac{{u}_{1}+{u}_{2}}{2},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{u}_{1},{u}_{2}\ge 0.$$
- The logarithmic mean:$$\begin{array}{ccc}\hfill L=L({u}_{1},{u}_{2})& =& \left(\right)open="\{"\; close>\begin{array}{cc}{u}_{1}\hfill & if{u}_{1}={u}_{2},\hfill \\ \frac{{u}_{2}-{u}_{1}}{ln{u}_{1},ln{u}_{2}}\hfill & if{u}_{1}\ne {u}_{2},\hfill \end{array}.{u}_{1},{u}_{2}0\hfill \end{array}$$
- The P-logarithmic mean:$$\begin{array}{ccc}\hfill {L}_{p}={L}_{p}({u}_{1},{u}_{2})& =& \left(\right)open="\{"\; close>\begin{array}{cc}{u}_{1}\hfill & if{u}_{1}={u}_{2},\hfill \\ {\left(\right)}^{\frac{{u}_{2}^{p+1}-{u}_{1}^{p+1}}{(p+1)({u}_{2}-{u}_{1})}}\frac{1}{p}\hfill \\ if{u}_{1}\ne {u}_{2},\hfill \end{array}\hfill & p\in \mathbf{R}\{-1,0\},{u}_{1},{u}_{2}0.\end{array}$$

- From Corollary 23,$$\begin{array}{c}|A({u}_{1}^{s},{u}_{2}^{s})-{L}_{s}^{s}({u}_{1},{u}_{2})|\hfill \\ \phantom{\rule{-20.54pt}{0ex}}\le \frac{{u}_{2}-{u}_{1}}{2}\left(\right)open="["\; close="]">\frac{({2}^{s+1}-1)(s+1)+(1-{2}^{s})(s+2)}{{2}^{s}(s+1)(s+2)}\left(\right)open="["\; close="]">\left(\right)open="|"\; close="|">\stackrel{\xb4}{h}\left({u}_{2}\right)+\left(\right)open="|"\; close="|">\stackrel{\xb4}{h}\left({u}_{1}\right)\hfill \\ +\frac{1}{2}\left(\right)open="|"\; close="|">b({u}_{1},{u}_{2},t)\end{array}.$$
- From Corollary 25,$$\begin{array}{c}\hfill |A({u}_{1}^{s},{u}_{2}^{s})-{L}_{s}^{s}({u}_{1},{u}_{2})|\le \frac{{u}_{2}-{u}_{1}}{2{(p+1)}^{\frac{1}{p}}}{\left(\right)}^{\frac{{\left(\right)}^{\stackrel{\xb4}{h}}q}{+}}s+1+\left(\right)open="|"\; close="|">b({u}_{1},{u}_{2},t)\\ \frac{1}{q}\end{array}$$

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Bector, C.R.; Singh, C. B-vex functions. J. Optim. Theory Appl.
**1991**, 71, 439–453. [Google Scholar] [CrossRef] - Chao, M.T.; Jian, J.B.; Liang, D.Y. Sub-b-convex functions and sub-b-convex programming. Oper. Res. Trans.
**2012**, 16, 1–8. [Google Scholar] - Hudzik, H.; Maligranda, L. Some remarks on s-convex functions. Aequ. Math.
**1994**, 48, 100–111. [Google Scholar] [CrossRef] - Orlicz, W. A note on modular spaces I. Bull. Acad. Polon. Sci. Ser. Math. Astronom. Phys.
**1961**, 9, 157–162. [Google Scholar] - Meftah, B. New integral inequalities for s-preinvex functions. Int. J. Nonlinear Anal.
**2017**, 8, 331–336. [Google Scholar] - Liao, J.; Tingsong, D. On Some Characterizations of Sub-b-s-Convex Functions. Filomat
**2016**, 30, 3885–3895. [Google Scholar] [CrossRef] - Ben-Israel, A.; Mond, B. What is invexity? ANZIAM J.
**1986**, 28, 1–9. [Google Scholar] [CrossRef] - Weir, T.; Mond, B. Pre-invex functions in multiple objective optimization. J. Math. Anal. Appl.
**1988**, 136, 29–38. [Google Scholar] [CrossRef] - Mohan, S.R.; Neogy, S.K. On invex sets and preinvex functions. J. Math. Anal. Appl.
**1995**, 189, 901–908. [Google Scholar] [CrossRef] - Suneja, S.K.; Singh, C.; Bector, C.R. Generalization of preinvex and B-vex functions. J. Optim. Theory Appl.
**1993**, 76, 577–587. [Google Scholar] [CrossRef] - Long, X.J.; Peng, J.W. Semi-B-preinvex functions. J. Optim. Theory Appl.
**2006**, 131, 301–305. [Google Scholar] [CrossRef] - Ahmad, I.; Jayswalb, A.; Kumarib, B. Characterizations of geodesic sub-b-s-convex functions on Riemannian manifolds. J. Nonlinear Sci. Appl.
**2018**, 11, 189–197. [Google Scholar] [CrossRef] [Green Version] - Boltyanski, V.; Martini, H.; Soltan, P.S. Excursions into Combinatorial Geometry; Springer Science & Business Media: Berlin/Heidelberg, Germany, 1997. [Google Scholar]
- Chen, X. Some properties of semi-E-convex functions. J. Math. Anal. Appl.
**2002**, 275, 251–262. [Google Scholar] [CrossRef] - Duca, D.I.; Lupsa, L. On the E-epigraph of an E-convex functions. J. Optim. Theory Appl.
**2006**, 129, 341–348. [Google Scholar] [CrossRef] - Dwilewicz, R.J. A short History of Convexity. Differ. Geom. Dyn. Syst.
**2009**, 11, 112–129. [Google Scholar] - Kiliçman, A.; Saleh, W. A note on starshaped sets in2-dimensional manifolds without conjugate points. J. Funct. Spaces
**2014**, 2014, 3. [Google Scholar] - Liu, W. New integral inequalities involving beta function via P-convexity. Miskolc Math. Notes
**2014**, 15, 585–591. [Google Scholar] - Martini, H.; Swanepoel, K.J. Generalized convexity notions and combinatorial geometry. Gongr. Numer.
**2003**, 164, 65–93. [Google Scholar] - Syau, Y.R.; Jia, L.; Lee, E.S. Generalizations of E-convex and B-vex functions. Comput. Math. Appl.
**2009**, 58, 711–716. [Google Scholar] [CrossRef] - Liao, J.; Tingsong, D. Optimality Conditions in Sub-(b, m)-Convex Programming. Univ. Politeh. Buchar. Sci. Bull.-Ser. A-Appl. Math. Phys.
**2017**, 79, 95–106. [Google Scholar] - Fok, H.; Von, S. Generalizations of some Hermite—Hadamard-type inequalities. Indian J. Pure Appl. Math.
**2015**, 46, 359–370. [Google Scholar] [CrossRef] - Noor, M.A.; Noor, K.I.; Iftikhar, S.; Safdar, F. Some Properties of Generalized Strongly Harmonic Convex Functions. Int. J. Anal. Appl.
**2018**, 16, 427–436. [Google Scholar] - Dragomir, S.S. On Hadamards inequality for convex functions on the co-ordinates in a rectangle from the plane. Taiwan. J. Math.
**2001**, 5, 775–788. [Google Scholar] [CrossRef] - Hua, J.; Xi, B.; Feng, Q. Inequalities of Hermite–Hadamard type involving an s-convex function with applications. Appl. Math. Comput.
**2014**, 246, 752–760. [Google Scholar] [CrossRef] - Dragomir, S.S.; Fitzpatrick, S. The Hadamard inequalities for s-convex functions in the second sense. Demonstr. Math.
**1999**, 32, 687–696. [Google Scholar] [CrossRef] - Barani, A.; Ghazanfari, A.G.; Dragomir, S.S. Hermite–Hadamard inequality for functions whose derivatives absolute values are preinvex. J. Inequal. Appl.
**2012**, 2012, 247. [Google Scholar] [CrossRef] - Pearce, C.E.M.; Pecaric, J. Inequalities for differentiable mappings with application to specialmeans and quadrature formulae. Appl. Math. Lett.
**2000**, 13, 51–55. [Google Scholar] [CrossRef]

© 2018 by the authors. 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**

Kiliçman, A.; Saleh, W.
Generalized Preinvex Functions and Their Applications. *Symmetry* **2018**, *10*, 493.
https://doi.org/10.3390/sym10100493

**AMA Style**

Kiliçman A, Saleh W.
Generalized Preinvex Functions and Their Applications. *Symmetry*. 2018; 10(10):493.
https://doi.org/10.3390/sym10100493

**Chicago/Turabian Style**

Kiliçman, Adem, and Wedad Saleh.
2018. "Generalized Preinvex Functions and Their Applications" *Symmetry* 10, no. 10: 493.
https://doi.org/10.3390/sym10100493