On Strongly m-convex Functions

Transcription

1 Matheatica Aeterna, Vol. 5, 205, no. 3, On Strongly -Convex Functions Teodoro Lara Departaento de Física y Mateáticas. Universidad de Los Andes. Venezuela. tlara@ula.ve Nelson Merentes Escuela de Mateáticas. Universidad Central de Venezuela. Caracas. Venezuela nerucv@gail.co Roy Quintero Departaento de Física y Mateáticas. Universidad de Los Andes. Venezuela. rquinter@ula.ve Edgar Rosales Departaento de Física y Mateáticas. Universidad de Los Andes. Venezuela. edgarr@ula.ve Abstract The ain purpose o this research is to introduce the deinition o a strongly -convex unction. To achieve this goal, we generalize the well known notion o a strongly convex unction by ollowing a siilar procedure to the one eployed to generate the notion o unctional - convexity ro the classical unctional convexity. Several properties o this new class o unctions are established as well as soe inequalities o Jensen type in the discrete case. In the course o this study, we prove soe additional and interesting results or the bigger class o -convex unctions. Matheatics Subject Classiication: 26A5, 52A30 Keywords: -convex unction, strongly convex unction, strongly - convex unction, Jensen type inequalities. This research has been partially supported by Central Bank o Venezuela.

2 522 Teodoro Lara, Nelson Merentes, Roy Quintero and Edgar Rosales Introduction The concepts o -convex and strongly convex unctions were introduced in [0] and [8], respectively. There are several papers [, 2, 4, 5, 0] in which we can ind soe results such as algebraic properties, inequalities o dierent type, aong others. In this research we cobine both deinitions in one, and we establish and prove soe properties or this type o unctions. Deinition. [, 2, 0] A unction : [0,b] R is called -convex, 0, i or any x, y [0,b] and t [0,] we have tx+ ty tx+ ty. Reark.2 It is iportant to point out that the above deinition is equivalent to tx+ ty tx+ ty, x,y and t as beore. Reark.3 I is an -convex unction, with [0, then we take x = y = 0 in and get 0 0. In the sae way, we ay deine the concept o a strongly convex unction. Deinition.4 [5] Let I R be an interval and c be a positive real nuber. A unction : I R is said to be strongly convex with odulus c i tx+ ty tx+ ty ct tx y 2, 2 with x,y I and t [0,]. Strongly convex unctions have been introduced by Polyak in [8]. Since strong convexity is a strengthening o the notion o convexity, soe properties o strongly convex unctions are just stronger versions o known properties o convex unctions. Strongly convex unctions have been used or proving the convergence o a gradient type algorith or iniizing a unction. They play an iportant role in optiization theory and atheatical econoics [7, 9]. Now we introduce a new deinition which cobine the two given above, let I [0,+, c be a positive real nuber and [0,]. Deinition.5 A unction : I R is called strongly -convex with odulus c i tx+ ty tx+ ty ct tx y 2, 3 with x,y I and t [0,]. Reark.6 Notice that or = the deinition o strongly convex unction is recasted. Moreover, i is strongly -convex with odulus c, then is strongly -convex with odulus k, or any constant 0 < k < c.

3 On Strongly -Convex Functions 523 Reark.7 Any strongly -convex unction is, in particular, -convex. However, there are -convex unctions, which are not strongly -convex with odulus c, or soe c > 0. Exaple.8 The unction : [0,+ R, given by x = 2 x4 5x 3 +9x 2 5x is 6-convex [6, ]; thereore, is -convex Lea 2, []. Nevertheless, 7 2 or any c >, is not strongly -convex with odulus c. In act, i were 3 2 strongly -convex with odulus c, then or all x,y 0 and t [0,], 2 tx+ 2 ty tx+ 2 ty c 2 t tx y2 ; in particular, by taking x =,y = 2 and t =, we get 2 contradicting the act that c > 3. 0 = c = 24 8 c, 2 Main Results In [4], soe basic properties or -convex unctions are proven. Now we state soe others results or the. Proposition 2. I, 2 : [0,b] R are -convex unctions, then the unction given by x = ax x [0,b] { x, 2 x} is also -convex. Proo. I x,y [0,b] and t [0,], we have and Whence tx+ ty t x+ t y tx+ ty 2 tx+ ty t 2 x+ t 2 y tx+ ty. tx+ ty = ax{ tx+ ty, 2 tx+ ty} tx+ ty. Proposition 2.2 I n : [0,b] R is a sequence o -convex unctions converging pointwise to a unction on [0,b], then is -convex.

4 524 Teodoro Lara, Nelson Merentes, Roy Quintero and Edgar Rosales Proo. I x,y [0,b] and t [0,], we have tx+ ty = li n n tx+ ty li n t n x+ t n y = tx+ ty. We establish now soe properties or strongly -convex unctions. Proposition 2.3 Let : [0,b] R be a strongly -convex unction with odulus c. I 0 n < <, then is strongly n-convex with odulus c. Proo. I n = 0, the proo is trivial ro Reark.3 ollows that is a starshaped unction i.e., tx tx or all x [0,b] and or all t [0,]. We consider now 0 < n < <. Let x,y [0,b] with y x. Then, n tx+n ty = tx+ t y n tx+ t y ct t x n 2 y n tx+ t y cnt t nx n 2 y tx+n ty cnt t x n 2. y Furtherore, since n <, we have x n y x y 0. Thus, x n y2 x y 2 and hence, tx+n ty tx+n ty cnt tx y 2. Now let ussee thecasex < y.asinreark.2, it isclear that3indeinition.5, is equivalent to tx+ ty tx+ ty ct tx y 2. 4 Thus, or x < y, n ntx+ ty = t x + ty n n 2 t x + ty ct t x y n 2 ntx+ ty cnt t n x y ntx+ ty cnt t n x y 2.

5 On Strongly -Convex Functions 525 Since n <, we have n x y x y < 0. Thereore, n x y2 x y 2 and thus, ntx+ ty ntx+ ty cnt tx y 2. Fro 4 and the previous inequality ollows that is a strongly n-convex unction with odulus c. Proposition 2.4 Let 2 and,g : [a,b] R, a 0. I is strongly -convex with odulus c and g is strongly 2 -convex with odulus c 2, then +g is strongly -convex with odulus c +c 2. Proo. Since g is strongly 2 -convex with odulus c 2 and 2, then by Proposition2.3, g isstrongly -convex withodulusc 2. Thus, orx,y [a,b] and t [0,], +gtx+ ty = tx+ ty+gtx+ ty tx+ ty c t tx y 2 +tgx+ tgy c 2 t tx y 2 = t +gx+ t +gy c +c 2 t tx y 2. Reark 2.5 According to Reark.6 and Proposition 2.4, +g is strongly -convex with odulus c and c 2. Proposition 2.6 I : [0,b] R is strongly -convex with odulus c and α > 0, then α is strongly -convex with odulus αc. In particular, i α, α is strongly -convex with odulus c. Proo. For x,y [0,b] and t [0,], αtx+ ty α[tx+ ty ct tx y 2 ] = tαx+ tαy αct tx y 2. Deinition 2.7 [2] Two unctions, g are said to be siilarly ordered on I i x ygx gy 0, 5 or all x,y I. Proposition 2.8 Let : [0,b] R be an -convex unction, g : [0,b ] R strongly -convex with odulus c and range [0,b ]. I g is nondecreasing and the unctions id and +id are siilarly ordered on [0,b], where id is the identity unction, then g is strongly -convex with odulus c.

6 526 Teodoro Lara, Nelson Merentes, Roy Quintero and Edgar Rosales Proo. Because is -convex, or all x,y [0,b] and t [0,], tx+ ty tx+ ty. Since g is nondecreasing and strongly -convex with odulus c, we have gtx+ ty gtx+ ty tgx+ tgy ct tx y 2. Since id and +id are siilarly ordered, by 5 we get [x x y y][x+x y+y] 0, or x y 2 x y 2. Indeed, g tx+ ty tg x+ tg y ct tx y 2. In [4], it was proved that i,g are both nonnegative, increasing and - convex unctions, then the product unction g is also -convex. However, in case o strong -convexity, this is not necessarily true. Exaple 2.9 Let c > 0 and 0,] be given. The quadratic unction : [0,b] R b > c deined by x x 2 is strongly -convex with odulus, but its square g = 2 is not strongly -convex with odulus c. Indeed, since the null unction is -convex, by Proposition 2.3, the unction is clearly strongly -convex with odulus, whence hal o the work is done. On the other hand, i g were a strongly -convex unction with odulus c, then inequality 3 would be true in particular or t 0 = [0,], x + 0 = 0, y 0 = ct 0 [0,b] and replaced by g. By calculating appropriately with the entioned values inequality 3 turns into c 2 t which is not true. So, gx = x4 or all x [0,b] cannot be strongly -convex with odulus c. Reark 2.0 Exaple 2.9 shows aong other aspects that the property o being strongly -convex is not inherited by the basic operation o ultiplication o unctions. Even ore, oregoing exaples proves that squaring a siple unction can destroy any possibility o conveying the property because the shown unction satisies the condition: or all 0,] is strongly -convex with odulus, but or all 0,] and or all c > 0 2 is not strongly -convex with odulus c. Now we give, or strongly -convex unctions, siilar results as Propositions 2. and 2.2.

7 On Strongly -Convex Functions 527 Proposition 2. I, 2 : [0,b] R are strongly -convex unctions with odulus c and c 2, then the unction given by x = ax{ x, 2 x} x [0,b] is strongly -convex with odulus c := in{c,c 2 }. Proo. I x,y [0,b] and t [0,], we have and Whence, tx+ ty t x+ t y c t tx y 2 tx+ ty ct tx y 2 2 tx+ ty t 2 x+ t 2 y c 2 t tx y 2 tx+ ty ct tx y 2. tx+ ty = ax{ tx+ ty, 2 tx+ ty} tx+ ty ct tx y 2. Proposition 2.2 I n : [0,b] R is a sequence o strongly -convex unctions with odulus c, converging pointwise to a unction on [0, b], then is strongly -convex with odulus c. Proo. It is siilar to the proo o Proposition 2.2. The next result perits to obtain a strongly -convex unction ro an -convex. Proposition 2.3 I : [0,b] R is an -convex unction and c any positive real constant, then the unction g deined by gx = x + cx 2 is strongly -convex with odulus c. Proo. I =, the result ollows ro Lea, [5]. I = 0, it is easy to veriy by using Reark.3 that g satisies deinition.5. I 0, and t 0, let x,y [0,b]. Then, gtx+ ty = tx+ ty+ctx+ ty 2 tx+ ty+ctx+ ty 2 = tx+cx 2 + ty+cy 2 ct tx 2 +2ct txy c t ty 2 x 2 t = tgx+ tgy ct t 2xy + y 2 t tgx+ tgy ct tx y 2.

8 528 Teodoro Lara, Nelson Merentes, Roy Quintero and Edgar Rosales Exaple 2.4 The unction : [0,+ R, given by x = ax+b is clearly -convex [0,] i b 0. Thereore, by Proposition 2.3, the unction gx = cx 2 +ax+b, b 0 is strongly -convex with odulus c > 0. Exaple 2.5 The unction : [0,+,0], given by x = lnx+ is -convex [2]. Thus, the unction gx = x 2 lnx+ is strongly -convex with odulus. Corollary 2.6 I : [0,b] R is strongly -convex with odulus c, then the unction h deined by hx = x+cx 2 is -convex. Proo. I is strongly -convex with odulus c, then is -convex. Hence, by the Proposition 2.3, h is strongly -convex with odulus c, and particularly, h is -convex. It is easy to see that i is -convex and c > 0, then x + c x2 is strongly -convex with odulus c, because ro Proposition 2.3, we have that x + c x2 is strongly -convex with odulus c, and by Reark.6, it is also odulus c. Nevertheless, this arguent is not applicable to check i x+cx 2 is strongly -convex with odulus c, even when it is. Proposition 2.7 I : [0,b] R is an -convex unction and c any positive real constant, then the unction g deined by gx = x + cx 2 is strongly -convex with odulus c. Proo. It is siilar to the proo o Proposition Discrete Jensen type inequalities In [3], it was proved a discrete Jensen type inequality or an -convex unction, by considering dierentiable. We give a siilar inequality without such hypothesis. Theore 3. Let t,...,t n > 0 and = t i. I : [0,+ R is an -convex unction, with 0,], then t i x i n i xi t i, or all x n i,...,x n [0,+.

9 On Strongly -Convex Functions 529 Proo. The proo is by induction on n. I n = 2, 2 t2 t i x i = x 2 + t x. Because o t = t 2 and the -convexity o, t2 x 2 + t x Thereore, 2 t i x i t t 2 x 2 + t x +t 2 x 2 x. = 2 2 i t i We assue now that the result is true or n, that is, n t i x i n n i xi t i, n i xi 2 i. and because x i [0,+ or i =,...,n, this inequality iplies n x i t i n n i xi t i. 6 n i Now, t i x i = = n t n x n + t i x i t n x n + But = t n and is -convex, thereore t i x i = t n x n + t n x n + By using 6 appropriately, t i x i t n x n + n n i t i n n n t i xi. t i xi xi n i t i xi. = n i t i xi n i.

10 530 Teodoro Lara, Nelson Merentes, Roy Quintero and Edgar Rosales Reark 3.2 I = = i.e., is a convex unction in Theore 3., t i x i t i x i, which is the classical discrete Jensen s inequality or a convex unction. We now present two versions o the discrete Jensen inequality or the class o strongly -convex unctions. Theore 3.3 Let t,...,t n be positive real nubers n 2 and = t i. I : [0,+ R is a strongly -convex unction with odulus c, with 0,], then t i x i or all x,...,x n [0,+. n i xi t i c n n i t i+ T i n i T i+ Proo. The proo is by induction on n. I n = 2, 2 t2 t i x i = x 2 + t x. Because o t = t 2 and the strong -convexity o, t2 x 2 + t x t 2 x 2 + t x ct t 2 Thereore, 2 t i x i 2 2 i t i = [t 2 x 2 +t x xi c 2 2 i Let us suppose that 7 holds or n, that is, n t i x i n n i t i c n 2 ] xi n i t i+ T i n i T i+ 2 2 x i+ i t k x k, T i x 2 x 2 k= 7 c [ t t 2 x 2 x 2 t i+ T i 2 i T i+ ]. 2 x i+ i t k x k. T i k= 2 x i+ i t k x k, T i k=

11 On Strongly -Convex Functions 53 and because x i [0,+ or i =,...,n, this inequality iplies n x i t i n n i xi t i 8 T n n i c n 2 2 t i+ T i x n i i+ i x k t k. T i+ T i k= Now, t i x i = t n x n + n t i xi. But = t n and is strongly -convex, thereore t i x i t n x n + n t i xi T n T n By using 8 appropriately, [ t i x i t n x n + Hence, ct n n c n 2 x n n n i t i t i+ T i n i T i+ ct n n t i x i n t n x n + n i t i n xi n i x i+ T i i x n xi n i n t i xi 2. k= t k x k 2 t i xi 2. c n 2 2 t i+ T i x n i i+ i t k x k ct n x T i+ T i n n 2 t i x i k= n = n i xi t i c n 2 t i+ T i x n i T n n i i+ i t k x k. T i+ T i k=

12 532 Teodoro Lara, Nelson Merentes, Roy Quintero and Edgar Rosales Beore the second version, soe basic notation is given and a lea is shown. Let 2 n N, 0 < b R and [0,] be given. Let x,...,x n [0,b] and t,...,t n > 0 with t + +t n =. Let us denote with the sybols a i and y i the ollowing real nubers: a i = t j i n 9 j=i y i = a i+ j=i+ j i+ t j x j i n. 0 Lea 3.4 Let x,...,x n [0,b] and t,...,t n > 0 with t + +t n = be given. Then the ollowing stateents are true:. For n = 2, t a 2 a x y 2 = t t 2 x x For n = k +, t a 2 a x y 2 = t t x k+ i 2 t i i=2 t x i For n = k +, let us set the ollowing two k-tuples {x i } k and {t i } k by: or all i =,...,k. x i = x i+ and t i = t i+ t I we denote by a i = k j=i t j and by y i = a i+ k j=i+ j i+ t j x j, then or all i =,...,k. a i = t a i+ and y i = y i+ Proo.. It ollows directly ater replacing the corresponding values o a,a 2 and y or n = Ater replacing the values o a,a 2 and y or n = k+ and by realizing that t = a By replacing the value o t j in the irst orula we get a i = k j=i t j+ = a i+ t t and by doing soething siilar with each pair t j,x j and using the previous orula or i+ we obtain y i = t a i+2 k j=i+ j i+ t j+ t x j+ = y i+.

13 On Strongly -Convex Functions 533 Reark 3.5 The k-tuple {t i } k deined above satisies: t + +t k =. Theore 3.6 Special case I : [0,b] R is strongly -convex with odulus c, then i t i x i n i t i x i c i t i a i+ a i x i y i 2 or all x,...,x n [0,b], t,...,t n > 0 with t + +t n = where a i and y i are deined by 9 and 0 and or all n 2. Proo. Let Pn be the proposition indicated by inequality. To prove Pn or all n 2 requires to check the veracity o the basic step P2. It is true because by being strongly -convex and by part ro Lea 3.4 ollows 2 i t i x i t x +t 2 x 2 ct t 2 x x 2 2 = 2 2 i t i x i c i t i a i+ a i x i y i 2. The inductive hypothesis is the stateent that Pk is true, that is, inequality is valid where k 2 is a natural nuber or all x,...,x k I, t,...,t k > 0 with t + +t k =. Now we have to prove that Pk + is also a true stateent. In act, given any two k+-tuples x,...,x k+ [0,b], t,...,t k+ > 0 with t + +t k+ =, by parts 2 and 3 ro Lea 3.4

14 534 Teodoro Lara, Nelson Merentes, Roy Quintero and Edgar Rosales and by being strongly -convex, it ollows that k+ k i t i x i = t x + t i t i x i k t x + t i t i x i [ k t x + t k+ k = i t i x i c k+ = i t i x i c k ct t x k i t i x i c c t a 2 a x y 2 k+ i=2 i+ t i+ a i+2 a i+ x i+ y i+ 2 c t a 2 a x y 2 i t i a i+ a i x i y i 2. i 2 t i t x i 2 i t i a i+ a i x i y i 2 This shows that Pk + ollows ro Pk. Consequently, by the principle o induction we can conclude that Pn is true or all n 2. Corollary 3.7 General case I : [0, b] R is strongly -convex with odulus c, then i t i x i i t i x i c n i t i a i+ x i y i 2 2 a i or all x,...,x n [0,b], t,...,t n > 0 with t + +t n = where a i and y i are deined by 9 and 0 and or all n 2. Fro[5, Theore4]andTheore3.6with = isplausible toestablish the ollowing conjecture: Conjecture 3.8 Given any two n-tuples x,...,x n [0,b], t,...,t n > 0 with t + +t n =, we believe that the ollowing orula is true t i x i x 2 = n t i a i+ a i x i y i 2 where x = t x + +t n x n, a i+ = a i t i and a i and y i are deined by 9 and 0 with =. ]

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