Inernaional Journal o Analysis and Applicaions Volume 16, Number 3 2018, 427-436 URL: hps://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-427 SOME PROPERTIES OF GENERALIZED STRONGLY HARMONIC CONVEX FUNCTIONS MUHAMMAD ASLAM NOOR, KHALIDA INAYAT NOOR, SABAH IFTIKHAR AND FARHAT SAFDAR Deparmen o Mahemaics, COMSATS Insiue o Inormaion Technology, Islamabad, Pakisan. Corresponding auhor: noormaslam@gmail.com Absrac. In his paper, we inroduce a new class o harmonic convex uncions wih respec o an arbirary riuncion F,, : K K [0, 1] R, which is called generalized srongly harmonic convex uncions. We sudy some basic properies o srongly harmonic convex uncions. We also discuss he suicien condiions o opimaliy or unconsrained and inequaliy consrained programming under he generalized harmonic convexiy. Several special cases are discussed as applicaions o our resuls. Ideas and echniques o his paper may moivae urher research in dieren ields. 1. Inroducion The concep o convexiy and generalized convexiy in he sudy o opimaliy o solve mahemaical programming, have been exended using innovaive ideas and echniques. For example, in earlier papers, Becor and Singh [3] inroduced a class o b-vex uncions. Chao e al. [4] considered new generalized sub-b-convex uncions and sub-b-convex ses. They proved he suicien condiions o opimaliy or boh unconsrained and inequaliy consrained sub-b-convex programming. Anderson e. al. [1] and Iscan [5] have invesigaed various properies o harmonic convex uncions. Noor and Noor [9] have shown ha he minimum o he diereniable harmonic convex uncions on he harmonic convex se can be characerized by a class o variaional inequaliies, which is called harmonic variaional inequaliy. To he bes o our Received 2018-01-19; acceped 2018-03-14; published 2018-05-02. 2010 Mahemaics Subjec Classiicaion. 26D15; 26D10; 90C23. Key words and phrases. harmonic convex uncions; F -harmonic convex uncions; opimaliy condiions. c 2018 Auhors reain he copyrighs o heir papers, and all open access aricles are disribued under he erms o he Creaive Commons Aribuion License. 427
In. J. Anal. Appl. 16 3 2018 428 knowledge, his ield is new one and has no developed as ye. A signiican class o convex uncions is ha o srongly harmonic convex uncions inroduced by Noor e. al. [11]. For recen applicaions, generalizaions and oher aspecs o convex and harmonic convex uncions, see [2 4,6,10,12 19] and reerences herein. Inspired by he research works [2 4, 6], we inroduce a new class o harmonic convex uncions wih respec o an arbirary uncion F.,.,., which is called srongly generalized harmonic convex uncion. One can show ha he generalized srongly harmonic convex uncions is quie general and uniied one. Several new and old classes o convex and harmonic convex uncions can be obained rom hese general harmonic convex uncions. We consider he suicien condiions o opimaliy or boh unconsrained and inequaliy consrained programming. Some properies o generalized srongly harmonic convex uncions and generalized srongly harmonic convex ses are discussed. Resuls obained in his paper may be considered as signiican improvemen o he known resuls. 2. Preliminaries In his Secion, we recall some basic conceps and resuls. We also inroduce some new conceps and discuss some special cases. Deiniion 2.1. [1]. A se K = [a, b] R n \ {0} is said o be a harmonic convex se, i K, x, y K, [0, 1]. We now consider he concep o generalized srongly harmonic convex uncions wih respec o an arbirary riuncion F,, : K K [0, 1] R. Deiniion 2.2. Le K is a nonempy harmonic convex se in R n \ {0}. A uncion : K R is said o be generalized srongly harmonic convex uncion on K wih respec o map F,, : K K [0, 1] R, i and only i, 1 x + y + F x, y,, x, y K, [0, 1]. 2.1 I. I F x, y, = 0 in Deiniion 2.2, hen i reduces o harmonic convex uncion. Deiniion 2.3. [5]. A uncion : K R, where K is a nonempy harmonic convex se R n \ {0}, is said o be a harmonic convex uncion on K, i and only i, 1 x + y, x, y K, [0, 1]. This shows ha every harmonic convex uncion is harmonic sub-f -convex uncion wih respec o he map F x, y, = 0, bu he converse may no be rue. See also [8 10, 17].
In. J. Anal. Appl. 16 3 2018 429 II. I F x, y, = 1 F x y in Deiniion 2.2, hen i reduces o F-srongly harmonic convex uncion Deiniion 2.4. [18]. A uncion : K R \ {0} R is said o be F-srongly harmonic convex uncion, i here exiss a non-negaive uncion F : X \ {0} [0,, such ha 1 x + y 1 F, x y x, y K, 0, 1. 2.2 I = 1 2, hen 2.2 reduces o 2 x + y x + y 2 1 4 F x y and he uncion is called F -srongly harmonic mid-convexharmonic Jensen-convex uncion. III. I F x, y, = c1 x y uncion wih modulus c > 0. Deiniion 2.5. in Deiniion 2.2, hen i reduces o srongly harmonic convex [11]. A uncion : K R, where K is a nonempy harmonic convex se in R n \ {0}, is said o be srongly harmonic convex uncion on K wih modulus c > 0, i and only i, 1 x + y c1 x y 2, x, y I, 0, 1. Theorem 2.1. [17]. Le K be a nonempy harmonic convex se in R n \ {0} and le : K R be diereniable on K. Then is harmonic quasi convex, i and only i, x y y, 0 x, y K. y x Deiniion 2.6. Le K is a nonempy harmonic convex se in R n \ {0}. A uncion : K R is said o be generalized srongly harmonic quasi convex uncion on K wih respec o map F,, : K K [0, 1] R, i and only i, max{x, y} + F x, y,, x, y K, [0, 1]. 2.3 I. I F x, y, = 0 in Deiniion 2.6, hen i reduces o harmonic quasi convex uncion. Deiniion 2.7. [20]. A uncion : K R, where K is a nonempy harmonic convex se R n \ {0}, is said o be a harmonic quasi convex uncion on K, i and only i, max{x, y}, x, y K. We now deine a new class o generalized srongly harmonic log-convex uncions. Deiniion 2.8. Le K is a nonempy harmonic convex se in R n \ {0}. A uncion : K R is said o be generalized srongly harmonic log-convex uncion on K wih respec o map F,, : K K [0, 1] R, i and only i, [x] 1 [y] + F x, y,, x, y K, [0, 1]. 2.4
In. J. Anal. Appl. 16 3 2018 430 From 2.4, i ollows ha [x] 1 [y] + F x, y, 1 x + y + F x, y, max{x, y} + F x, y,, x, y K, [0, 1], This shows ha every generalized srongly harmonic log-convex uncion is generalized srongly harmonic convex and every generalized srongly harmonic convex uncion is generalized srongly harmonic quasi convex, bu he converse is no rue. 3. Main Resuls In his secion, we inroduce he concep o generalized harmonic convex se and harmonic convex uncions wih respec o an arbirary map F,, and discuss is properies. Theorem 3.1. I i : K R, i = 1, 2, 3,..., m are generalized srongly harmonic convex uncions wih respec o map F i : K K [0, 1] R, respecively. Then he uncion = a i i, a 1 0, i = 1, 2, 3,..., m, is generalized srongly harmonic convex wih respec o F = m a ib i. Proo. For all x, y K and [0, 1], we have = a i i a i [1 i x + i y + F i x, y, ] = 1 a i i x + a i i y + a i F i x, y, = 1 x + y + F x, y,. Theorem 3.2. I he uncion i : K R are generalized srongly harmonic convex uncions wih respec o F i x, y, respecively. Then = max{ i, i = 1, 2, 3,..., m} is also generalized srongly harmonic convex wih respec o F = max{f i }.
In. J. Anal. Appl. 16 3 2018 431 Proo. Consider, = max { i }, i = 1, 2, 3,..., m 1 max{ i x} + max{ i y} + max{f i x, y, = 1 x + y + F x, y,. So, x is harmonic sub-f convex uncion wih respec o F x, y, = max{ i, i = 1, 2, 3,..., m}. Theorem 3.3. I he uncion : K R is a generalized srongly harmonic convex uncion wih respec o F x, y, and g : R R is a linear uncion, hen g is generalized srongly harmonic convex wih respec o F = g F. Proo. Since is generalized srongly harmonic convex uncion wih respec o F x, y, and g is a an increasing uncion, i ollows ha g = g g1 x + y + F x, y, = 1 gx + gy + gf x, y, = 1 g + g + g F x, y,. Tha is, g is generalized srongly harmonic convex uncion wih respec o F = g F and his complees he proo. The ollowing heorem gives a necessary and suicien characerizaion o a diereniable generalized srongly harmonic convex uncion wih respec o a map F,,. Theorem 3.4. Le K be a harmonic convex se. I : K R is a diereniable generalized srongly harmonic convex uncion on he harmonic convex se K wih respec o he map F x, y,, hen 1 x, 2 x y, x y y x + lim 0 F x,y, +, x, y K. x y lim 0 F x,y, + + lim F y,x, 0 +, x, y K. Proo. 1. Le be a generalized srongly harmonic convex uncion. Then 1 x + y + F x, y,, which can be wrien as y x + F x, y, y+x y x.
In. J. Anal. Appl. 16 3 2018 432 Since is diereniable uncion, so aking he limi in he above inequaliy, as 0, we have which is 1. Changing he role o x and y, in 3.1, we obain Adding 3.1 and 3.2, we have F x, y, y x + lim x,, 3.1 x y F y, x, x y + lim y,. 3.2 y x x y, x y lim F x, y, F y, x, + lim, which is he required 2. This complees he proo. Deiniion 3.1. [17]. A uncion : K R, where K is a nonempy harmonic convex se R n \ {0} is said o be a harmonic quasi convex uncion, i, or each x, y K wih x y, we have y, equivalenly, i y, Deiniion 3.2. y x > 0, hen x > y. y x 0; or [17]. A uncion : K R, where K is a nonempy harmonic convex se R n \ {0} is said o be a harmonic pseudo-convex uncion, i, or each x, y K wih y, or equivalenly, i x < y, hen y, y x < 0. y x 0, we have x y; Theorem 3.5. Le K be a harmonic convex se and : K R be a diereniable generalized srongly harmonic convex uncion on he harmonic convex se K wih respec o he map F x, y,. I lim 0 F x,y, x y, x, y K, hen is harmonic quasi-convex. Furhermore, i lim 0 F x,y, x, y K, hen is harmonic pseudo-convex. < x y, Proo. Suppose ha x y, or any x, y K and 0, 1. Then rom Theorem 3.4, we have I lim 0 F y,x, F y, x, y, x y + lim. y x x y, hen x y + lim 0 F x,y, rom Deiniion 3.1, we have is harmonic quasi-convex uncion. Similarly, i x < y, we also have x, pseudo-convex uncion. 0. So, y, y x 0. Thereore, x y < 0. So, rom he Deiniion 3.2, we have is harmonic Now, we are going o inroduce a new concep o generalized harmonic convex se. Deiniion 3.3. Le K s R n+1 \{0} be a nonempy se. A se K s is said o be generalized harmonic convex wih respec o F x, y, : R n R n [0, 1] R, i, 1 α + β + F x, y, K s, x, α, y, β K s,
In. J. Anal. Appl. 16 3 2018 433 or all x, y R n, and [0, 1]. We now invesigae some characerizaions o generalized srongly harmonic convex uncion : K R in erm o heir epigraph E. Deiniion 3.4. [2]. Le K be a nonempy in R n and le : K R be a uncion. Then epigraph o, denoed by E is deined by E = {x, α : x K, α R, x α}. Theorem 3.6. A uncion : K R is generalized srongly harmonic convex wih respec o F x, y, : R n R n [0, 1] R, i and only i, epigraph o is generalized harmonic convex se wih respec o F,,. Proo. Suppose ha is harmonic sub-f-convex uncion wih respec o F x, y,. Le x, α and y, β E. Then x, y K, x α and y β, we have 1 x + y + F x, y, 1 α + β + F x, y,. Hence rom Deiniion, one has, 1 α + β + F x, y, Thus E is generalized harmonic convex se wih respec o F. E Conversely, assume ha E is generalized harmonic convex se wih respec o F. Le x, y K, hen x, x and y, y belong o E. Thus or [0, 1],, 1 x + y + F x, y, This urher ollows ha, E. 1 x + y + F x, y, Tha is is a generalized srongly harmonic convex uncion wih respec o F.,.,.. Theorem 3.7. I K si is a amily o generalized harmonic convex se wih respec o a map F x, y,, hen i I K si is a generalized harmonic convex se wih respec o F x, y,. Proo. Le x, α, y, β i I K si, [0, 1]. Then or each i I, x, α, y, β I i. Since K si is a generalized harmonic convex se wih respec o F.,.,., i ollows ha, 1 x + y + F x, y, K si, i I.
In. J. Anal. Appl. 16 3 2018 434 Thus, 1 x + y + F x, y, i I K si. Hence, i I K si is generalized harmonic convex se wih respec o F x, y,. We apply he above resuls o he nonlinear programming problem. Firs, we consider he unconsrain problem. Theorem 3.8. Le : K R be diereniable and generalized srongly harmonic convex uncion wih respec o map F,,. Consider he opimal problem min{x x K}. I x K and he relaion x, holds or each x K, hen x is he opimal soluion o on K. x x lim F x, x, 0, 3.3 Proo. For any x k, rom Theorem 3.4, we have x, x x lim F x, x, x x. 3.4 From 3.3 and 3.4, we have x x 0. So, x is an opimal soluion o on K. Nex, we apply he above resuls o he nonlinear programming problem wih inequaliy consrains: min x P g s. g i x 0, i I = {1, 2, 3,..., m} x R n Denoe he easible se o P g by S g = {x R n g i x 0, i I}. Deiniion 3.5. Le K be a nonempy harmonic convex se in R n. The uncion : K R is said o be generalized srongly harmonic pseudo convex on K wih respec o F : K K [0, 1] R, i or each x, y K and 0, 1, rom y, y x + lim 0 F y,x, 0 one can have x y. Theorem 3.9. Karush-Kuhn-Tucker Suicien Condiions The uncion is diereniable generalized srongly harmonic pseudo convex wih respec o F.,.,. : K K [0, 1] R, g i x i I are diereniable and generalized srongly harmonic convex wih respec o F : K K [0, 1] R. Assume ha x S g is a KKT poin o P g, ha is, here exis muliplier λ i 0 i I such ha x, x x + i I x λ i g x, x x = 0, λ i g x, = 0. 3.5 x x
In. J. Anal. Appl. 16 3 2018 435 I Then x is an opimal soluion o he problem P g. F x, x, lim F x, x, λ i lim, x S g. 3.6 0 0 i I Proo. For any x S g, we have g i x 0, g i x = 0, i I x = {i I g i x = 0}. Thereore, rom he generalized srongly harmonic convexiy o g i x and Theorem 3.4, we obain g x, x x lim F x, x, 0, or i I x. 3.7 From 3.5, one has x, x x = i I x λ i g x, x x. In view o 3.6 and rom 3.7, we have x, x x + lim F x, x, i I x = 0. i I x λ i g x, λ i [ g x, x x + i I x λ i F x, x, lim ] x x lim F x, x, So, x, x x + lim F x, x, 0. From he generalized srongly harmonic pseudo convexiy o x, we have x x. Thereore x is an opimal soluion o he problem P g. Acknowledgemens The auhors would like o hank he Recor, COMSATS Insiue o Inormaion Technology, Pakisan, or providing excellen research and academic environmens. Reerences [1] G. D. Anderson, M. K. Vamanamurhy and M. Vuorinen, Generalized convexiy and inequaliies, J. Mah. Anal. Appl., 3352007, 1294-1308. [2] M. S. Bazaraa, D. Hani, C. M. Shey, Nonlinear Programming Theory and Algorihms John Wiley and Sons, New York, 1993. [3] C.R. Becor, C. Singh, B-vex uncions, J. Opim. Theory Appl., 71 1991, 439-453. [4] M. T. Chao, J. B. Jian and D. Y. Liang, Sub-b-convex uncions and sub-f-convex programming, Oper. Res. Trans., 162012, 1-8.
In. J. Anal. Appl. 16 3 2018 436 [5] I. Iscan, Hermie-Hadamard ype inequaliies or harmonically convex uncions. Hace, J. Mah. Sas., 4362014, 935-942. [6] J. Liao and T. Du, On some characerizaions o sub-b-s-convex uncions, Filoma, 30142016, 3885-3895. [7] C. P. Niculescu and L. E. Persson, Convex Funcions and Their Applicaions, Springer-Verlag, New York, 2006. [8] M. A. Noor, Some developmens in general variaional inequaliies, Appl. Mah. Compu. 2512004, 199-277. [9] M. A. Noor and K. I. Noor, Harmonic variaional inequaliies, Appl. Mah. In. Sci., 1052016, 1811-1814. [10] M. A. Noor and K. I. Noor, Some implici mehods rsolving harmonic variaional inequalies, Iner. J. Anal. Appl. 1212016, 10-14. [11] M. A. Noor, K. I. Noor and S. Iikhar, Hermie-Hadamard inequaliies or srongly harmonic convex uncions, J. Inequ. Special Func.,732016, 99-113. [12] M. A. Noor, K. I. Noor and S. Iikhar, Inegral inequaliies or diereniable relaive harmonic preinvex uncionssurvey, TWMS J. Pure Appl. Mah. 712016, 3-19. [13] M. A. Noor, K. I. Noor and S. Iikhar, Hermie-Hadamard inequaliies or harmonic preinvex uncions, Saussurea 612016, 34-53. [14] M. A. Noor, K. I. Noor and S. Iikhar, Inegral inequaliies o Hermie-Hadamard ype or harmonic h, s-convex uncions, In. J. Anal. Appl., 1112016, 61-69. [15] M. A. Noor, K. I. Noor, S. Iikhar and F. Sadar, Inegral inequaliies or relaive harmonic s, η-convex uncions, Appl. Mah. Compu. Sci. 112016, 27-34. [16] M. A. Noor, K. I. Noor, S. Iikhar and C. Ionescu, Hermie-Hadamard inequaliies or co-ordinaed harmonic convex uncions, U.P.B. Sci. Bull., Ser: A, 7912017, 25-34. [17] M. A. Noor, K. I. Noor and S. Iikhar, Some characerizaions o harmonic convex uncions, In. J. Anal. Appl. 1522017, 179-187. [18] M. A. Noor, B. Bin-Mohsin, K. I. Noor and S. Iikhar, Relaive srongly harmonic convex uncions and heir characerizaions, J. Nonlinear Sci. Appl. in press. [19] J. Pecaric, F. Proschan, and Y. L. Tong, Convex Funcions, Parial Orderings and Saisical Applicaions, Acdemic Press, New York, 1992. [20] T.-Y. Zhang, Ai-P. Ji, F. Qi, Inegral inequaliies o Hermie-Hadamard ype or harmonically quasi-convex uncions, Proc. Jangjeon. Mah. Soc., 1632013, 399-407.