MERCER’S INEQUALITY FOR h-CONVEX FUNCTIONS
arXiv:1801.01775v1 [math.GM] 19 Dec 2017
M.W. ALOMARI Abstract. A generalization of Mercer inequality for h-convex function is presented. As application, a weighted generalization of triangle inequality is given.
1. Introduction The class of h-convex functions, which generalizes convex, s-convex (denoted by Ks2 , [1]), Godunova-Levin functions (denoted by Q(I), [3]) and P -functions (denoted by P (I), [9]), was introduced by Varoˇsanec in [10]. Namely, for real intervals I and J, the h-convex function is defined as a non-negative function f : I → R which satisfies f (tα + (1 − t) β) ≤ h (t) f (α) + h (1 − t) f (β) , where h : J → R is a non-negative function defined on J, such that t ∈ (0, 1) ⊆ J ⊆ (0, ∞) and x, y ∈ I. Accordingly, some properties of h-convex functions were discussed in the same work of Varoˇsanec. The famous references about these classes are [1]–[4] and [7]–[9]. Let w1 , w2 , · · · , wn be positive real numbers (n ≥ 2) and h : J → R be a non-negative supermultiplicative function. In [10], Varoˇsanec discussed the case that, if f is a non-negative h-convex on I, then for x1 , x2 , · · · , xn ∈ I the following inequality holds ! n n X wk 1 X h (1.1) wk xk ≤ f (xk ), f Wn Wn k=1
k=1
where Wn =
n P
wk . If h is submultiplicative function and f is an h-concave then inequality is reversed. In
k=1
case h(t) = t we refer to the classical version of Jensen’s inequality. n If f is convex on I, then for any finite positive increasing sequence (xk )k=1 ∈ I, we have ! n n X X (1.2) wk f (xk ), wk xk ≤ f (x1 ) + f (xn ) − f x1 + xn − k=1
k=1
where w1 , w2 , · · · , wn are positive real numbers such that
n P
wk = 1. This inequality was established by
k=1
Mercer in [6] and it is considered as a variant of Jensen’s inequality. In this work, a generalization of Mercer inequality for h-convex function is presented. As application, a weighted generalization of triangle inequality is given. 2. Mercer analogue inequality for h-convex functions In order to prove our main result, we need the following Lemma which generalizes Lemma 1.3 in [6]. Lemma 1. Let h : J → R be a non-negative supermultiplicative function on J. Let α, β ∈ [0, 1] such that α + β = 1 and h (α) + h (β) ≤ 1. For any h-convex function f defined on a real interval I and finite positive n increasing sequence (xk )k=1 ∈ I, we have (2.1)
f (x1 + xn − xk ) ≤ f (x1 ) + f (xn ) − f (xk )
(1 ≤ k ≤ n).
If h is submultiplicative function, h (α) + h (β) ≥ 1 for all α, β ∈ [0, 1] with α + β = 1 and f is an h-concave then inequality (2.1) is reversed. Date: January 8, 2018. 2000 Mathematics Subject Classification. 26D15. Key words and phrases. h-Convex function, Mercer inequality, Jensen inequality. 1
2
M.W. ALOMARI
Proof. Let 0 < x1 ≤ · · · ≤ xn and α, β ∈ [0, 1] such that α + β = 1 with h (α) + h (β) ≤ 1. Following Mercer approach in [6]. Let us write yk = x1 + xn − xk . Then x1 + xn = yk + xk , so that the pairs x1 , xn and xk , yk possess the same midpoint. Since that is the case there exists α, β ∈ [0, 1] such that xk = αx1 + βxn and yk = βx1 + αxn , where α + β = 1 and 1 ≤ k ≤ n. Employing the h-convexity of f we get f (yk ) = f (βx1 + αxn ) ≤ h (β) f (x1 ) + h (α) f (xn ) ≤ (1 − h (α)) f (x1 ) + (1 − h (β)) f (xn ) = f (x1 ) + f (xn ) − [h (α) f (x1 ) + h (β) f (xn )] ≤ f (x1 ) + f (xn ) − f (αx1 + βxn ) = f (x1 ) + f (xn ) − f (αx1 + βxn ) = f (x1 ) + f (xn ) − f (xk ) , and this proves the required result.
Now, we are ready to state our main result. Theorem 1. Let h : J → R be a non-negative supermultiplicative function on J. Let w1 , w2 , · · · , wn be n n P P wk h W wk and positive real numbers (n ≥ 2) such that Wn = ≤ 1. If f is h-convex on I, then for n k=1
k=1
any finite positive increasing sequence (xk )nk=1 ∈ I, we have ! n n X wk 1 X h (2.2) wk xk ≤ f (x1 ) + f (xn ) − f (xk ). f x1 + xn − Wn Wn k=1 k=1 n P wk ≥ 1 and f is an h-concave then inequality (2.2) is reversed. If h is submultiplicative function, h W n k=1
Proof. Since f
1 Wn
n P
wk = 1, we have
k=1
n 1 X x1 + xn − wk xk Wn k=1
!
! n X wk =f (x1 + xn − xk ) Wn k=1 n X wk f (x1 + xn − xk ) by (1.1) ≤ h Wn k=1 n X wk ≤ h [f (x1 ) + f (xn ) − f (xk )] by (2.1) Wn k=1 X n n X wk wk = [f (x1 ) + f (xn )] h − h f (xk ) Wn Wn k=1 k=1 n X wk f (xk ) by assumption ≤ f (x1 ) + f (xn ) − h Wn k=1
and this proves the result in (2.2).
One of the direct application and interesting benefit of (2.2) is to offer an upper bound for the converse of h-Jensen inequality (1.1), by rearranging the terms in (2.2) we get ! n n X 1 X wk (2.3) f (xk ) ≤ f (x1 ) + f (xn ) − f x1 + xn − wk xk . h Wn Wn k=1
k=1
For instance, if f (x) = |x|, h(t) = t and Wn = 1, then we have the following refinement of the celebrated triangle inequality which is of great interests itself n n X X wk |xk | ≤ |x1 | + |xn | − x1 + xn − wk xk . (2.4) k=1
k=1
MERCER’S INEQUALITY FOR h-CONVEX FUNCTIONS
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This inequality can be generalized for norms by considering the mapping f (x) = kxk (x ∈ L), where L is a linear space. References [1] W.W. Breckner, Stetigkeitsaussagen f¨ ur eine Klasse verallgemeinerter konvexer funktionen in topologischen linearen R¨ aumen, Publ. Inst. Math., 23 (1978), 13–20. [2] S.S. Dragomir, J. Peˇ cari´ c and L.E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21 (1995) 335–341. [3] E.K. Godunova and V.I. Levin, Neravenstva dlja funkcii ˇsirokogo klassa, soderˇ zaˇsˇ cego vypuklye, monotonnye i nekotorye drugie vidy funkcii, Vyˇ cislitel. Mat. i. Mat. Fiz. Mevuzov. Sb. Nauc. Trudov, MGPI, Moskva, 1985, 138–142. [4] H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Math., 48 (1994), 100–111. [5] J. Jensen, Sur les fonctions convexes et les in´ egalit´ es entre les valeurs moyennes, Acta Math., 30 (1906), 175–193. [6] A.McD. Mercer, A variant of Jensen’s inequality, JIPAM, 4 (4) Article 73, 2003. [7] D.S. Mitrinovi´ c and J. Peˇ cari´ c, Note on a class of functions of Godunova and Levin, C. R. Math. Rep. Acad. Sci. Can., 12 (1990), 33–36. [8] D.S. Mitrinovi´ c, J. Peˇ cari´ c and A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic, Dordrecht, 1993. [9] C.E.M. Pearce and A.M. Rubinov, P -functions, quasi-convex functions and Hadamard-type inequalities, J. Math. Anal. Appl., 240 (1999), 92–104. [10] S. Varoˇsanec, On h-convexity, J. Math. Anal. Appl., 326 (2007), 303–11. Department of Mathematics, Faculty of Science and Information Technology, Irbid National University, 2600 Irbid 21110, Jordan. E-mail address:
[email protected]
