SOME CHARACTERIZATIONS OF HARMONIC CONVEX FUNCTIONS

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1 International Journal o Analysis and Applications ISSN Volume 15, Number , DOI: / SOME CHARACTERIZATIONS OF HARMONIC CONVEX FUNCTIONS MUHAMMAD ASLAM NOOR, KHALIDA INAYAT NOOR, SABAH IFTIKHAR Abstract. In this paper, we show that the harmonic convex unctions have some nice properties, which convex unctions enjoy. We also discuss some basic properties o harmonic convex unctions. The techniques and ideas o this paper may be a starting point or uture research. 1. Introduction Convexity theory played an important and undamental role in the developments o various branches o engineering, inancial mathematics, economics and optimization. In recent years, the concept o convex unctions and its variant orms have been extended and generalized using innovative techniques to study complicated problems. It is well known that convexity is closely related to inequality theory. The optimality conditions o dierentiable convex unctions are characterized by variational inequalities, the origin o which can be traced back to Euler, Lagrange and Newton. On the other hand, convex unctions are related to integral inequalities. which are called the Harmite-Hadamard type integral inequalities. For recent developments, see [1, 3, 9, 16] and reerence therein. Related to the arithmetic means, we have harmonic means. The harmonic means have applications in electrical circuit theory and other branches o sciences. For example, the total resistance o a set o parallel resistors is obtained by adding up the reciprocal o the individual resistance value and then considering the reciprocal o their total. Also the harmonic means are used in developing parallel algorithms or solving various problems, see Noor [5]. A signiicant class o convex unctions, called harmonic convex was introduced by Anderson et al. [1] and Iscan [3], independently. Noor and Noor [6, 7] have shown that the optimality conditions o the dierentiable harmonic convex unctions on the harmonic convex set can be expressed by a class o variational inequalities, which is called the harmonic variational inequality. For recent developments and applications, see [5 7, 9 14]. To the best o our knowledge, this ield is new one and has not been developed as yet. In this paper, we show that the harmonic convex unctions have some nice properties [2], which convex unctions enjoy. We have investigated several basic properties o harmonic convex unctions. We obtained the necessary and suicient characterization o a dierentiable harmonic convex and harmonic quasi convex unctions. It is worth mentioning that the harmonic variational inequalities is a new class o variational inequalities and is not studied much. The interested readers are encouraged to study these harmonic variational inequalities. It is high time to ind the applications o these inequalities along with eicient numerical methods. 2. Preliminaries First o all, we recall the ollowing basic concepts. Deinition 2.1. [1]. A set K R n \ {0} is said to be a harmonic convex set, i K, x, y K, t [0, 1]. tx + 1 ty Received 4 th August, 2017; accepted 10 th October, 2017; published 1 st November, Mathematics Subject Classiication. 26D15; 26D10; 90C23. Key words and phrases. Convex unctions; general preinvex unctions; dierentiability; Hermite-Hadamard inequality. c 2017 Authors retain the copyrights o their papers, and all open access articles are distributed under the terms o the Creative Commons Attribution License. 179

2 180 NOOR ET AL. Deinition 2.2. [3]. A unction : K R, where K is a nonempty harmonic convex set in R n \{0}. The unction is said to be a harmonic convex unction on K, i and only i, 1 tx + ty, x, y K, t [0, 1]. tx + 1 ty The unction is called strictly harmonic convex on K, i the above inequality is true as a strict inequality or each distinct x and y K and or each t 0, 1. The unction : K R is called harmonic concave strictly harmonic concave on K i is harmonic convex strictly harmonic convex on K. Remark 2.1. Geometric Interpretation We now consider the geometric interpretation o harmonic convex unction. Let x and y be two distinct points in the domain o, and consider the point tx+1 ty, with t 0, 1. Note that 1 tx+ty gives the weighted arithmetic mean o x and y, while tx+1 ty gives the value o at the point tx+1 ty. So, or a harmonic convex unction, the value o at the points on the path tx+1 ty whose initial point is x and terminal point is y, is less than or equal to the chord joining the points x, x and y, y. For a harmonic concave unction, the chord is on or below the unction itsel. Deinition 2.3. [16]. A unction : K R, where K is a nonempty harmonic convex set in R n \{0}. The unction is said to be a harmonic quasi convex unction on K, i and only i, max{x, y}, x, y K, t [0, 1]. tx + 1 ty The unction is said to be harmonic quasi concave, i is harmonic quasi convex. A unction is harmonic quasi convex, i whenever y x, y is greater than or equal to all the values o at the points o the path tx+1 ty. A unction is said to be strictly harmonic quasi convex, i strict inequality holds or x y. Deinition 2.4. [9]. A unction : K R, where K is a harmonic convex set in R n \ {0}. The unction is said to be a harmonic log-convex unction on K, i and only i, [x] 1 t [y] t, x, y K, t [0, 1]. 2.1 tx + 1 ty From 2.1, it ollows that tx + 1 ty [x] 1 t [y] t 1 tx + ty max{x, y}, x, y K, t [0, 1], This shows that, every harmonic log-convex unction is harmonic convex and every harmonic convex unction is harmonic quasi convex, but the converse is not true. Also rom 2.1, we have log 1 t log x + t log y, tx + 1 ty x, y K, t [0, 1]. For the properties and inequalities o harmonic log-convex unctions, see [9, 13, 14]. 3. Main Results In this section, we discuss some properties o harmonic convex unction and harmonic quasi convex unction. Theorem 3.1. I K i is a amily o harmonic convex set, then i I K is a harmonic convex set.

3 CHARACTERIZATIONS OF HARMONIC CONVEX FUNCTIONS 181 Proo. Let x, y i I K i, t [0, 1]. Then, or each i I, x, y K i, since K i is a harmonic convex set, it ollows that tx + 1 ty K i, i I. Thus Hence, i I K i is harmonic convex set. tx + 1 ty i IK i. Theorem 3.2. Let K be a harmonic set. I i : K R, i = 1, 2, 3,..., m are harmonic convex unctions. Then the unction = a i i, a i 0, i = 1, 2, 3,..., m, is harmonic convex unction. Proo. Let K be a harmonic set. Then x, y K and t [0, 1], we have = a i i tx + 1 ty tx + 1 ty a i [1 t i x + t i y] This shows that is a harmonic convex unction. = 1 t a i i x + t a i i y = 1 tx + ty. Theorem 3.3. Let K be a harmonic set. I the unctions i : K R are harmonic convex unctions, then = max{ i, i = 1, 2, 3,..., m} is also harmonic convex. Proo. Consider, tx + 1 ty } = max { i, i = 1, 2, 3,..., m tx + 1 ty = w tx + 1 ty 1 t w x + t w y = 1 t max{ i x} + t max{ i y} = 1 tx + ty. This implies that x is harmonic convex unction. Theorem 3.4. Let K be a harmonic set. I the unction : K R is harmonic convex unction and g : R R is a linear unction, then g is a harmonic convex unction. Proo. Let be a harmonic convex unction and g be a linear unction. Then g = g tx + 1 ty tx + 1 ty g1 tx + ty This shows that g is a harmonic convex unction. = 1 tgx + tgy = 1 tg x + tg y.

4 182 NOOR ET AL. Theorem 3.5. Let K be a harmonic set. I the unction : K R is a harmonic quasi convex unction and g : R R is a nondecreasing unction, then g is a harmonic quasi convex unction. Proo. Let be a harmonic quasi convex unction and g be a nondecreasing unction. Then g tx + 1 ty g[max{x, y}] = max{g x, g y}, which implies that g is a harmonic quasi convex unction. Theorem 3.6. Let : K R be a harmonic convex unction. I µ = in x K x, then the set E = {x K : x = µ} is a harmonic convex set. I is strictly harmonic convex unction, then E is a singleton. Proo. Let x, y E. Since is harmonic convex unction, so 1 tx + ty = µ, tx + 1 ty which implies that tx+1 ty E and hence E is a harmonic convex set. For the second part, assume that x = y = µ. Since K is a harmonic convex set, so or t 0, 1, tx+1 ty K. Further, since is strictly harmonic convex unction, so < 1 tx + ty = µ. tx + 1 ty This contradicts that µ = in x K x and hence E is singleton. Lemma 3.1. Let K be a nonempty harmonic convex set in R n \ {0} and let : K R be a harmonic convex unction. Then the level set is a harmonic convex se. Proo. Let x, y K α. Then x α, y α. Consider, Hence α = {x K : x α, α R}, 3.1 tx + 1 ty 1 tx + ty 1 tα + tα = α. tx+1 ty K α and thereore K α is a harmonic convex set. Remark 3.1. We would like to mention that the converse o above result is not true. Deinition 3.1. [2]. Let K be a nonempty in R n and let : K R be a unction. Then epigraph o, denoted by E, is deined by E = {x, α : x K, α R, x α}. Theorem 3.7. Let K be a nonempty harmonic convex set in R n \ {0} and let : K R. Then is harmonic convex, i and only i, E is a harmonic convex set. Proo. Assume that is harmonic convex unction and let x, α, y, β E. Then x α and y β. Consider, or t [0, 1], tx + 1 ty 1 tx + ty 1 tα + tβ.

5 CHARACTERIZATIONS OF HARMONIC CONVEX FUNCTIONS 183 Thus tx+1 ty, 1 tα + tβ E and hence E is a harmonic convex set. Conversely, assume that E is a harmonic convex set and let x, y K. Then x, x, y, y E and by the harmonic convexity o E, we have, 1 tx + ty E. tx + 1 ty Thus, tx+1 ty 1 tx + ty or each t 0, 1, that is, is harmonic convex unction. Remark 3.2. I the unction : K R is harmonic log-convex unction on the harmonic convex set K, then E is a harmonic convex set. Theorem 3.8. Let K be a harmonic convex set and let : K R be a harmonic convex unction. Then any local minimum o is a global minimum. Proo. Let x K be a local minimum o the harmonic convex unction.. Assume the contrary, that is, y < x, or some y K. Since is harmonic convex unction, we have 1 tx + ty x, y K, t [0, 1]. tx + 1 ty Thus tx + 1 ty x t[], rom which it ollows that or some small t > 0, < x. tx + 1 ty Contradicting the local minimum. Hence every local minimum o is a global minimum. Strictly harmonic quasi convex and strictly harmonic quasi concave unctions are especially important in nonlinear programming because they ensure that a local minimum and a local maximum over a harmonic convex set is a global minimum and a global maximum, respectively. Theorem 3.9. Let K be a harmonic convex set and let : K R be a strictly harmonic quasi convex unction. Consider the problem o minimum o x subject to x K. I x is a local optimal solution, then x is also a global optimal solution. Proo. Assume, on the contrary, that there exists an ˆx K with ˆx < x. xˆx convexity o K, t x+1 tˆx K or each t 0, 1. Since x is a local minimum by assumption, xˆx x < t 0, 1. t x + 1 tˆx But because is strictly harmonic quasi convex and ˆx < x, we have xˆx < x t 0, 1. t x + 1 tˆx This contradicts the local optimality o x, and the proo is complete. By the harmonic Lemma 3.2. Let K be a nonempty harmonic convex set in R n \ {0}. Then a unction : K R is a harmonic quasi convex unction, i and only i, the level set K α deined by 3.1 is a harmonic convex set. Proo. Suppose that is harmonic quasi-convex unction and let x, y K α. Thereore x, y K and max{x, y} α. Also x 1 = tx+1 ty K, since K is a harmonic convex set. Using the harmonic quasi-convexity o, we have max{x, y} α. tx + 1 ty

6 184 NOOR ET AL. Hence x 1 K α and thereore K α is harmonic convex set. Conversely, suppose that K α is harmonic convex set or each real number α. Let x, y K α. Furthermore, let t 0, 1 and x 1 = tx+1 ty. Note that x, y K α or α = max{x, y}. By assumption, K α is harmonic convex set, so that x 1 K α. Thereore α = max{x, y}. tx + 1 ty Hence, is harmonic quasi-convex and the proo is complete. Remark 3.3. The level set deined as K α = {x K : x α, α R}, is sometimes reerred to as a lower-level set, to dierentiate it rom the upper-level set deined as K α = {x K : x α, α R}, which is harmonic convex set, i and only i, is harmonic quasi concave. Theorem I the unction : K R is harmonic convex unction such that x < y, or all x, y K, then is strictly harmonic quasi convex unction. Proo. By the harmonic convexity o, we have 1 tx + ty tx + 1 ty < x; since x < y, which shows that the harmonic convex unction is strictly harmonic quasi convex unction. Deinition 3.2. The unction : K R is said to be harmonic pseudo convex unction with respect to a strictly positive unction b, such that y < x < x + tt 1bx, y, x, y K, t 0, 1. tx + 1 ty Theorem I the unction : K R is harmonic convex unction such that y < x, then is harmonic pseudo convex unction with respect to a strictly positive unction b,. Proo. Let y < x and let be a harmonic convex unction. Then tx + 1 ty x + t < x + t1 t where bx, y = x y > 0, the result result. = x + tt 1x y < x + tt 1by, x, Remark 3.4. I the unction : K R is harmonic log-convex unction such that y < x, then the harmonic log-convex unction is harmonic pseudo convex under the same sense. Theorem Let K be a nonempty harmonic convex set in R n \ {0} and let : K R be strictly harmonic quasi convex unction and lower semicontinuous. Then is harmonic quasi convex unction. Proo. Let x, y K. I x y, then by the strict harmonic quasi convexity o, we must have < max{x, y} t 0, 1. tx + 1 ty Now, suppose that x = y. To show that is harmonic quasi convex, we need to show that y t 0, 1. tx + 1 ty By contradiction, suppose that > y or some t 0, 1. tx + 1 ty

7 tx+1 ty CHARACTERIZATIONS OF HARMONIC CONVEX FUNCTIONS 185 Denote by x. Since is lower semicontinuous, there exists a t 0, 1 such that x > > y = x. 3.2 t x + 1 ty Hence by the strict harmonic quasi convexity o and since t x+1 ty > y, we have x < t x+1 ty, contradicting 3.2. This completes the proo. The ollowing theorem gives a necessary and suicient characterization o a dierentiable harmonic quasi convex unction. Theorem Let K be a nonempty harmonic convex set in R n \ {0} and let : K R be dierentiable on K. Then is harmonic quasi convex, i and only i, x y y, 0, x, y K. Proo. Let be harmonic quasi convex and let x, y K be such that Using the Taylor series, we have = y + t y, x + t x y. where α [ y; t ] 0, as t 0. By the harmonic quasi convexity o, we have implies that t y, + t [ ] α y; t. 1 tx+ty y and hence the above equation [ ] + t α y; t 0. Dividing by t and taking the limit in the above inequality as t 0, we have y, 0. Conversely, suppose that x, y K and that x y. We need to show that tx+1 ty y, or all x K and t 0, 1. We do this by showing that the set L = {x : x = tx+1 ty, t 0, 1, x > y} is empty. By contradiction, suppose that there exists an x L. Thereore x = tx+1 ty or some t 0, 1 and x > y. Since is dierentiable, it is continuous and there must exits a δ 0, 1, such that x y 1 µx > y or each µ δ, 1, + µy and x > x y 1 δx +δy. By this inequality and the mean value theorem, we must have 0 < x x y 1 δx = 1 t x y ˆx,, δy where ˆx = x y 1 µ x +µ y or some µ δ, 1. From it is clear that ˆx > y. Dividing 3.3 by x y 1 t > 0, it ollows that ˆx, > 0, which in turn implies that ˆx, > But on the other hand ˆx > y x, and ˆx is harmonic combination o x and y. assumption o the theorem ˆxx ˆx, ˆx x 0, and thus we must have 0 ˆx,. The above inequality is not compatible with 3.4. Thereore, L is empty, and the proo is complete. By the

8 186 NOOR ET AL. The ollowing theorem was derived by Noor and Noor [6, 7] and it gives a necessary and suicient characterization o a dierentiable harmonic convex unction. Theorem [6]. Let K be a harmonic convex set. I : K R is a dierentiable harmonic convex unction on the harmonic convex set K, then 1 x, x y, x, y K 2 x y, 0, x, y K, x y but the converse is not true. Theorem [6]. Let be a dierentiable harmonic convex unction on the harmonic convex set K. Then x K is a minimum o, i and only i, x K satisies x, 0, y K. x y Remark 3.5. The inequality o the type x, x y 0 is known as harmonic variational inequality, which was introduced by Noor and Noor [6, 7]. Deinition 3.3. A unction : K R, where K is a nonempty harmonic set R n \ {0}. The unction is said to be a harmonic pseudo-convex unction, i or each x, y K with y, 0, we have x y; or equivalently, i x < y, then y, < 0. Deinition 3.4. A unction : K R, where K is a nonempty harmonic set R n \ {0}. The unction is said to be a harmonic quasi convex unction, i or each x, y K with x y, we have y, 0; or equivalently, i y, > 0, then x > y. Theorem Let K be a harmonic convex set and : K R a dierentiable harmonic convex unction on the harmonic convex set K. I x y, x, y K, then is harmonic quasi-convex unction. Furthermore, i x < y, x, y K, then is harmonic pseudo-convex unction. Proo. Let be dierentiable harmonic convex unction on the harmonic convex set K. Then rom Theorem 3.14, we have y, x y. 0. Thereore, rom Theorem 3.13, we have is harmonic quasi- I x y, then y, convex unction. Similarly, i x < y, we also have y, harmonic pseudo-convex unction. Acknowledgements < 0. So, rom the Deinition 3.3, we have is The authors would like to thank Rector, COMSATS Institute o Inormation Technology, Pakistan, or providing excellent research and academic environments. Reerences [1] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Generalized convexity and inequalities, J. Math. Anal. Appl., , [2] M. S. Bazaraa, D. Hani, C. M. Shetty, et al. Nonlinear Programming Theory and Algorithms Second Edition[M]. The United States o America: John Wiley and Sons, [3] I. Iscan, Hermite-Hadamard type inequalities or harmonically convex unctions. Hacet. J. Math. Stats., , [4] C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications, Springer-Verlag, New York, [5] M. A. Noor, Advanced Convex Analysis and Optimization, Lecture Notes, CIIT, [6] M. A. Noor and K. I. Noor, Harmonic variational inequalities, Appl. Math. In. Sci., , [7] M. A. Noor and K. I. Noor, Some implicit methods or solving harmonic variational inequalities, Inter. J. Anal. App , [8] M. A. Noor and K. I. Noor, Auxiliary principle techinique or variational inequalities, Appl. Math. In. Sci., , [9] M. A. Noor, K. I. Noor and M. U. Awan. Some characterizations o harmonically log-convex unctions. Proc. Jangjeon. Math. Soc., ,

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