Refinement of Steffensen s Inequality for Superquadratic functions

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1 Int. Journal of Math. Analysis, Vol. 8, 14, no. 13, HIKARI Ltd, Refinement of Steffensen s Inequality for Superquadratic functions Mohammed Muniru Iddrisu Department of Mathematics University for Development Studie P. O. Box 4 Navrongo, Ghana Christopher A. Okpoti Department of Mathematics University of Education Winneba, Ghana Kazeem A. Gbolagade Department of Computer Science University for Development Studies P. O. Box 4 Navrongo, Ghana Copyright c 14 Mohammed Muniru Iddrisu, Christopher A. Okpoti and Kazeem A. Gbolagade. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we discuss superquadratic functions and some results related to Steffensen s inequality. Particularly, we state and prove the superquadratic form of the refined Steffensen s inequality. Applications of the new results are established. Mathematics Subject Classification: 6D1, 6D15 Keywords: Steffensen s inequality, superquadratic functions, Convex functions, refinement

2 61 M.M. Iddrisu, C.A. Okpoti and K.A. Gbolagade 1 Introduction The concept of superquadratic functions was first introduced by S. Abramovich et al. in [1] and [] and it has since been dealt with in numerous papers (see for example [3]),[4]and [5]. The definition of a superquadratic function is a simple modification of the geometrical notion of a convex function. In the case of a superquadratic function, it is required that ϕ lies above its tangent line plus a translation of ϕ itself. Our task in this paper is to present some refinements of the Steffensen s inequality b b λ f(t)dt b a g(t)f(t)dt a+λ a f(t)dt, (1) where λ = b a g(t)dt, f and g are integrable functions defined on (a, b), f is decreasing and for each t (a, b), g(t) 1 (see also [6], [7]), [8] and [9]. Preliminary Notes We give some definitions here. Definition.1 (Convex functions) Let I be an interval in R. Then ψ : I R is said to be convex if for all t 1,t I and for all positive λ and μ satisfying λ + μ =1, we have ψ(λt 1 + μt ) λψ(t 1 )+μψ(t ). () Geometrically, a convex function is defined as ψ(t ) ψ(t 1 )+C t1 (t t 1 ) where C t1 is a slope for each t 1 I and t I. [Note: If ψ is differentiable at t 1 then C t1 = ψ (t 1 ).] A function ψ is said to be concave if ψ is convex (i.e. if the inequality () is reversed). If it is strict for all t 1 t, ψ is said to be strictly concave. Some examples of convex functions are: t, t k for k>1 and t k for <k<1, e t, t(log t) k for k 1, log t, etc. Concave functions are t k for <k<1, log t, t for t etc. Definition. A function ϕ :[, ) Ris superquadratic provided that for all t there exists a constant C t1 Rsuch that for all t. ϕ(t ) ϕ(t 1 ) ϕ( t t 1 ) C t1 (t t 1 ) (3) }{{} extra term

3 Refinement of Steffensen s inequality 613 The absolute values in the definition of superquadratic functions are employed instead of extending ϕ :[,b) Rto be an even function. If ϕ(u) =u, we have the identity v u (v u) =u(v u) where C u = u. We observe that if ϕ(u) is superquadratic and a, b then ϕ(u) (au+b) is also superquadratic. Any function ϕ(u) satisfying ϕ(u) 1 for all u is superquadratic. Some examples of superquadratic functions are u p for p, u p for p<, ϕ(u) =u log u for u> and ϕ() = (See also [1],[],[3]). 3 Results and Discussion In [8], J.E. Pecaric gave a refinement of inequality (1) as ( p λ f(t)g(t)dt) f(t) p dt (4) where λ = ( g(t)dt) p, f :[, 1] R is a nonnegative and nonincreasing function, g :[, 1] R is an integrable function with g(t) 1( t [, 1]) and p 1. Putting p = 1 and replace f(t) with ψ (t) in (4), we obtain the following theorem: Theorem 3.1 Let the function g : [, 1] R be continuous such that g(t) 1. If ψ :[, 1] R is a convex, differentiable function with ψ() =. Then ( ) ψ g(t)dt g(t)ψ (t)dt (5) for all t [, 1]. Proof Let ψ (t) denote the differential of ψ(t) which is increasing and therefore ψ (t) is nonincreasing for all t [, 1]. Then by (4) we have λ g(t)ψ (t)dt ψ (t)dt This simplifies to λ ψ (t)dt g(t)ψ (t)dt Thus ( ) ψ g(t)dt g(t)ψ (t)dt Let us also establish a refinement of (5), in this case, by considering a superquadratic function. We first give a lemma.

4 614 M.M. Iddrisu, C.A. Okpoti and K.A. Gbolagade Lemma 3. Suppose that ϕ is a superquadratic function and that its differential exists. Then for all t 1, there exists ϕ (t ),ϕ (t 1 ) Rsuch that for all t, t 1 t. ϕ (t ) ϕ (t 1 ) ϕ( t 1 t ) t 1 t Proof By definition., if ϕ is superquadratic, then ϕ(t 1 ) ϕ(t )+ϕ (t )(t 1 t )+ϕ( t 1 t ). (6) for all t 1,t. Interchanging t 1 and t, we have ϕ(t ) ϕ(t 1 )+ϕ (t 1 )(t t 1 )+ϕ( t t 1 ) (7) Adding inequalities (6) and (7) we obtain ϕ (t )(t 1 t )+ϕ (t 1 )(t t 1 )+ϕ( t 1 t ) or [ϕ (t ) ϕ (t 1 )](t 1 t )+ϕ( t 1 t ) Assume that t 1 >t, then ϕ (t ) ϕ (t 1 ) ϕ( t 1 t ) t 1 t. Theorem 3.3 Let g :[, 1] R, be an integrable function such that g(t) 1, ( t [, 1]). If ϕ :[, 1] Ris superquadratic and differentiable with ϕ() =, then ( ) ϕ g(t)dt + g(t) ϕ( t gdt ) t gdt dt g(t)ϕ (t)dt. Proof Let G(t) = t f(x)dx t. The function g(t) is continuous and the differential of G(t) denoted G (t) =g(t). Let ( t ) F (t) =ϕ{g(t)} = ϕ f(x)dx. (8) Now differentiating (8) and integrating the result, we obtain F (1) = F (t)dt = g(t)ϕ (G(t))dt. ( ) ϕ g(t)dt = g(t)ϕ (G(t))dt. (9)

5 Refinement of Steffensen s inequality 615 Apply Lemma 3. by letting t = G(t) and t 1 = t, then ϕ (G(t)) ϕ (t) Substituting (1) into (9), we obtain ( ) ( 1 ϕ g(t)dt g(t) ϕ (t) ϕ( t G(t) ). (1) t G(t) ) ϕ( t G(t) ) dt t G(t) Therefore ( ) ϕ( t G(t) ) 1 ϕ g(t)dt + g(t) dt g(t)ϕ (t)dt t G(t) as required. Remark 3.4 By choosing ϕ(u) =u p, for p, Theorem 3.3 becomes ( p g(t)dt) + ( t G(t) )p g(t) dt t G(t) }{{} extra term p g(t)t (p 1) dt. (11) Remark 3.5 Consider p =in Remark 3.4 and write {G (t)} =G(t)G (t) = g(t)g(t) since G (t) =g(t). Then the extra term becomes g(t)[t G(t)]dt = = = tg(t)dt tg(t)dt tg(t)dt ( g(t)g(t)dt {G (t)} dt g(t)dt). Thus inequality (11) becomes ( ( g(t)dt) + tg(t)dt g(t)dt) = tg(t)dt. Therefore, equality is attained for p =. Applications. For n 1, let g(t) = 1 (1 + ε sin nt) for ε = 1 or 1. Then ( π ) ϕ g(t)dt = ϕ(π), π g(t)ϕ (t)dt = ϕ(π) nε π ϕ(t) cos(nt)dt

6 616 M.M. Iddrisu, C.A. Okpoti and K.A. Gbolagade Also, let g(t) = 1 (1 + ε cos nt) for ε = 1 or 1. Then π If ϕ is superquadratic, then g(t)ϕ (t)dt = ϕ(π) (1 + ε)+ nε π ϕ(t) sin(nt)dt. 1. π ϕ( t G(t) ) ϕ(π)+ (1 + ε sin nt) dt ϕ(π) nε π ϕ(t) cos(nt)dt (1) t G(t). π ϕ(π)+ (1+ε cos nt) ϕ( t G(t) ) t G(t) dt φ(π) (1+ε)+ nε π φ(t) sin(nt)dt. (13) Remark 3.6 Using Remark 3.5, If ϕ(t) =t, the extra term for (1) is π (1 + ε sin nt)(t G(t))dt = and the extra term for (13) is π = π πε n t(1 + ε sin nt)dt π π (1 + ε cos nt)(t G(t))dt = π = π t(1 + ε cos nt)dt π Thus, (1) attains equality π + π πε n and (13) also attains equality =π nε ( ) 4π n π +π π =π (1 + ε)+ nε ( ) 4π n for ε = 1 or 1.

7 Refinement of Steffensen s inequality Conclusion A refinement of the Steffensen s inequality is thus presented. As well, superquadratic functions are discussed and established for the new Steffensen s inequality (5). This led to the applications of the results in superquadratic form. Acknowledgements. The first author wish to thank Frederic SYMESAK and the French Government for the attention and financial assistance during his research period at the University of Angers, France. References [1] S. Abramovich, G. Jameson and G. Sinnamon, Refining Jensen s Inequality, Bull. Math. Soc. Sci. Math. Roumanie (N.S). 47(95)(1-) (4) [] S. Abramovich, G. Jameson and G. Sinnamon, Inequalities For Averages of Convex and Superquadratic Functions, J. Inequal Pure Appl. Math 5(4) (4), article 91. [3] S. Abramovich, On Superquadracity, J. Math. Inequality, 3(3) (9), [4] S. Abramovich, S. Ivelic and J.E. Pecaric, Improvement of Jensen- Steffensen s Inequality for Superquadratic functions, Banach J. Math. Anal. bf 4(1) (1), [5] J. A. Oguntuase, L.- E. Persson, E. K. Essel and B. A. Popoola, multidimensional Hardy-Type Inequalities Via Superquadracity, Banach J. Math. Anal. () (8), [6] D.S. Mitrinovic, J.E. Pecaric and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, (1993) [7] D.S. Mitrinovic and J.E. Pecaric, On the Bellman Generalization of Steffensen s Inequality III, J. Math. Anal. and Appl., 135 (1988), [8] J.E. Pecaric, On the Bellman Generalization of Steffensen s Inequality, J. Math. Anal. and Appl., 88 (198), [9] J.E. Pecaric, On the Bellman Generalization of Steffensen s Inequality II, J. Math. Anal. and Appl., 14 (1984), Received: February 17, 14