GEOMETRICAL PROOF OF NEW STEFFENSEN S INEQUALITY AND APPLICATIONS

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1 Available online at Adv. Inequal. Appl. 214, 214:23 ISSN: GEOMETRICAL PROOF OF NEW STEFFENSEN S INEQUALITY AND APPLICATIONS MOHAMMED M. IDDRISU 1,, CHRISTOPHER A. OKPOTI 2, KAZEEM A. GBOLAGADE 3 1 Department of Mathematics, University for Development Studies, P.O. Box 24, Navrongo, Ghana 2 Department of Mathematics, University of Education, Winneba, Ghana 3 Department of Computer Science, University for Development Studies, P.O. Box 24, Navrongo, Ghana Copyright c 214 Iddrisu, Okpoti Gbolagade. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we give a geometrical proof of a new Steffensen s inequality for convex functions. In addition, we present applications of the Steffensen s inequality leading to the determination of Fourier coefficients. Keywords: Steffensen s inequality, geometrical proof, Fourier coefficient. 21 AMS Subject Classification: 26D Introduction The the following inequality was discovered in 1918 by Steffensen [9] (1) b b λ g(s)ds b a g(s) f (s)ds a+λ a g(s)ds, where λ = b a f (s)ds, f g are integrable functions defined on (a,b), g is monotone decreasing for each s (a,b), f (s) 1; see also [5], [8], [7] [6] the references therein. Godunova Levin in [3] noted that the generalisation of (1) by Bellman in [2] was incorrect. Corresponding author address: muiinr@yahoo.com(m.m. Iddrisu) Received February 9, 214 1

2 2 M.M. IDDRISU, C.A. OKPOTI, K.A. GBOLAGADE Pecaric [8] corrected the Bellman generalisation with a narrow subclass. The corrected result is (2) p λ f (s)g(s)ds) g(s) p ds, where λ = ( ) 1 p, f (s)ds g : [,1] R is a nonnegative nonincreasing function, f : [,1] R is an integrable function with f (s) 1 ( s [,1]) p 1, for the proof; see [8] the references therein. The purpose of this paper is to present a refinement of inequality (2) with proofs consisting of both analytical geometrical. 2. Preliminaries We begin with convex functions. Definition 2.1. (Convex functions) Let I be an interval in R. Then ψ : I R is said to be convex if for all t 1,t 2 I for all positive λ µ satisfying λ + µ = 1, we have (3) ψ(λt 1 + µt 2 ) λψ(t 1 ) + µψ(t 2 ). A convex function necessarily is continuous for t 1,t 2 I. A function ψ is said to be strictly convex if for all t 1 t 2, ψ is said to be strictly convex. Remark 2.1. The convexity of a function ψ : I R means geometrically that, the function ψ falls below (or lies on not above) the chord joining the endpoints (t 1,ψ(t 1 )) (t 2,ψ(t 2 )), for every t 1,t 2 I. Intuitively, a convex function has a tangent line at each point lies above of its tangent lines. That is, for each t I there exists a slope C t such that ψ(s) ψ(t) +C t (s t), x I. We remark here that if ψ is differentiable at t then C t = ψ (t). Definition 2.2. A function ψ is said to be concave if ψ is convex (i.e. if the inequality (3) is reversed). If it is strict for all t 1 t 2, ψ is said to be strictly concave.

3 GEOMETRICAL PROOF OF NEW STEFFENSEN S INEQUALITY AND APPLICATIONS 3 Remark 2.2. If ψ (t) exists at each point of the interval I, then a necessary sufficient condition that ψ(t) is convex is that ψ (t) for all t I. For the above discussion, we refer authors to [5] [1]. Some examples of convex functions are: t, t k for k > 1 t k for < k < 1, e t, t logt, logt concave functions are: t k for < k < 1, logt, t for t so on. 3. Main results We first present a refinement of inequality (2) here. Theorem 3.1. Let the function f : [,1] R be continuous such that f (s) 1. If ψ : [,1] R is a convex, differentiable function with ψ() =, then ) (4) ψ f (s)ds f (s)ψ (s)ds for all s [,1]. Proof. Let p = 1. Since the differential of ψ(s) denoted ψ (s) is increasing ψ (s) is nonincreasing for all s [,1], substitution of g(s) = ψ (s) in (2) gives λ f (s)ψ (s)ds ψ (s)ds. This simplifies to λ ψ (s)ds f (s)ψ (s)ds, (5) ψ(λ) ψ() Since λ = f (s)ds ψ() =, thus (5) becomes This completes the proof. f (s)ψ (s)ds. ) ψ f (s)ds f (s)ψ (s)ds Let us consider a case of a simple function f on an interval [s,s 2 ] such that s < s 2 1. We give some definitions

4 4 M.M. IDDRISU, C.A. OKPOTI, K.A. GBOLAGADE Definition 3.1. Let a 1 a 2 be real numbers. Define a function f : [s,s 2 ] R by a 1 if s s < s 1, f (s) = a 2 if s 1 s s 2. Then f is called a simple function since for every s [s,s 2 ], we have f (s) = a j for j = 1,2. Let us obtain a continuous function f ε from f. Let ε >, we have the partition {[s,s 1 ε),[s 1 ε,s 1 + ε),[s 1 + ε,s 2 ]} of [s,s 2 ]. Definition 3.2. Let a 1 a 2 be real numbers. Define a function f ε : [s,s 2 ] R by a 1 if s s < s 1 ε, f ε (s) = a 2 a 1 2ε (s s 1 + ε) + a 1 if s 1 ε s < s 1 + ε, a 2 if s 1 + ε s s 2. Remark 3.1. Let us remark that f ε is continuous in [s,s 2 ] since lim s s f ε (s) = f ε (s ) for every s [s,s 2 ]. f(s) f ε(s) a 1 a 1 A 1 a 2 a 2 A 2 s F igure1. s 1 s 2 s s F igure2. s 1 ε s 1 s 1 + ε s2 s Lemma 3.1. Let f (s) f ε (s) be functions as in Definitions respectively. Then (6) s f (s)ds = s f ε (s)ds.

5 GEOMETRICAL PROOF OF NEW STEFFENSEN S INEQUALITY AND APPLICATIONS 5 Proof. The midpoint of the line f ε (s) = a 2 a 1 (s s 1 + ε) + a 1 f or s 1 ε s < s 1 + ε 2ε is P = ( s 1, a ) 1+a 2 2. (See Figure 2). Therefore, the areas A 1 = ε [ ( )] a1 + a 2 a 1 = ε (a 1 a 2 ), Therefore, we have A 2 = ε [( ) ] a1 + a 2 a 2 = ε (a 1 a 2 ). A 1 = A 2. Lemma 3.2. Let f (s) f ε (s) be functions as in Definitions respectively. If ψ(s) is a convex, differentiable function with ψ() =, then [ f ε (s) f (s)]ψ (s)ds = a 1 a 2 s 2ε s1 +ε [ψ(s) ψ(s 1 )]ds. Proof. Write (7) s [ f ε (s) f (s)]ψ (s)ds = s f ε (s)ψ (s)ds f (s)ψ (s)ds. s The second term on the right side of (7) gives s f (s)ψ (s)ds = s1 s a 1 ψ (s)ds + s 1 a 2 ψ (s)ds, (8) s f (s)ψ (s)ds = (a 1 a 2 )ψ(s 1 ) + a 2 ψ(s 2 ) a 1 ψ(s ). Also, the first term on the right side of (7) is expressed as s f ε (s)ψ (s)ds = + + s1 ε s s1 +ε s 1 +ε a 1 ψ (s)ds [ a2 a 1 (s s 1 + ε) + a 1 ]ψ (x)ds 2ε a 2 ψ (s)ds.

6 6 M.M. IDDRISU, C.A. OKPOTI, K.A. GBOLAGADE Applying integration by parts, we obtain f ε (s)ψ (s)ds = a 1 [ψ(s 1 ε) ψ(s )] + a [ 2 a s1 ] +ε 1 2εψ(s 1 + ε) ψ(s)ds s 2ε which simplifies to (9) + a 1 ψ(s 1 + ε) a 1 ψ(s 1 ε) + a 2 ψ(s 2 ) a 2 ψ(s 1 + ε), f ε (s)ψ (s)ds = a 1 a 2 s 2ε s1 +ε Thus, the difference between inequalities (8) (9) gives ψ(s)ds + a 2 ψ(s 2 ) a 1 ψ(s ). as required. [ f ε (s) f (s)]ψ (s)ds = (a 1 a 2 ) s 2ε s1 +ε [ψ(s) ψ(s 1 )]ds Lemma 3.3. Let g(s) be a continuous function on the interval [s,s 2 ]. Then lim η 1 s1 +η g(s)ds = g(s 1 ). 2η s 1 η Proof. Let η > set I(η) = 1 s1 +η g(s)ds. 2η s 1 η Continuity of g at s 1. Let ε >, there exists δ > such that g(s) g(s 1 ) < ε whenever s s 1 < δ. Since for η < δ, we have I(η) g(s 1 ) 1 s1 +η g(s) g(s 1 ) ds 2η s 1 η s 1 η (s 1 δ,s 1 + δ) s 1 + η (s 1 δ,s 1 + δ). Thus, s s 1 < η hence I(η) g(s 1 ) < ε. Therefore I(η) g(s 1 ) as η. Lemma 3.4. Let f be a simple function defined as in Definition 3.1 such that f 1. If ψ is a convex, differentiable function with ψ() =, then ( ) ψ f (s)ds f (s)ψ (s)ds. s s

7 GEOMETRICAL PROOF OF NEW STEFFENSEN S INEQUALITY AND APPLICATIONS 7 Proof. Let f ε (s) be continuous as in Definition 3.2. Then by Lemma 3.1, Theorem 3.1 Lemma 3.2 respectively, we have ( ) ( ψ f (s)ds = ψ s s s ) f ε (s)ds f ε (s)ψ (s)ds s f (s)ψ (s)ds + Thus, by Lemma 3.3, when ε, we obtain as required. s f (s)ψ (s)ds + (a 1 a 2 ) s 2ε [ f ε (s) f (s)]ψ (s)ds s1 +ε ( ) ψ f (s)ds f (s)ψ (s)ds s s [ψ(s) ψ(s 1 )]ds. Theorem 3.1. Let f be a simple function on [,1] such that f (s) 1 for all s [,1]. If ψ is a convex, differentiable function with ψ() =, then ) ψ f (s)ds f (s)ψ (s)ds. Proof. Let f be a simple function. There exists { = s,s 1,,s n = 1} {a 1,a 2,,a n } such that f (s) = a j on [s j,s j+1 ) for j n 1. Let < ε < min s j+1 s j define f ε (s) = f (s) if And if s [,s 1 ε) [s 1 + ε,s 2 ε) [s j + ε,s j+1 ε) [s n 1 + ε,1). f ε (s) = a ( j+1) a j (s s j + ε) + a j 2ε s [s j ε,s j + ε)

8 8 M.M. IDDRISU, C.A. OKPOTI, K.A. GBOLAGADE where j = 1,,n 1. (See Figure 3 Figure 4). Then, following Lemma 3.4, we have Therefore as required. f (s)ds = ) ) ψ f (s)ds = ψ f ε (s)ds f ε (s)ds f (s)ψ n 1 a j a j+1 (s)ds + j=1 2ε s j +ε s j ε ) ψ f (s)ds f (s)ψ (s)ds [ψ(s) ψ(s j )]ds. f(s) f(s) a 1 a 1 a 2 a 2.. a n 1 a n 1 a n a n s s 1 s 2 s n 1 s n s s s 1 s 2 s n 1 s n s F igure3 F igure4 4. Applications In Theorem 3.1, replace 1 by a >. Thus ( 2π ) ψ f (s)ds 2π f (s)ψ (s)ds. We estimate the Fourier coefficients of ψ : a n = 1 2π ψ(s) cos(ns)ds 2π b n = 1 2π ψ(s) sin(ns)ds 2π

9 GEOMETRICAL PROOF OF NEW STEFFENSEN S INEQUALITY AND APPLICATIONS 9 for n 1. For the estimate of b n, let f (s) = 1 2 (1 + ε cosns) for ε = 1 or 1. Thus 1 2π (1 + ε cosns)ds = π 2 Hence 1 2π (1 + ε cosns)ψ (s)ds = ψ(2π) (1 + ε) + nε ψ(π) ψ(2π) (1 + ε) + nε 2 2 2π 2π ψ(s) sin(ns)ds ψ(s) sin(ns)ds Take ε = 1 or 1 we obtain ψ(π) ψ(2π) nπ b n ψ(π) nπ. Also, for the estimate of a n, let f (s) = 2 1 (1 + ε sinns) for ε = 1 or 1. Thus 1 2π (1 + ε sinns)ds = π 2 1 2π (1 + ε sinns)ψ (s)ds = ψ(2π) nε π ψ(s) cos(ns)ds. Hence ψ(π) a n ψ(2π) 2 nε 2 2π ψ(s) cos(ns)ds. Take ε = 1 or 1 we obtain ψ(π) ψ(2π)/2 nπ a n ψ(2π)/2 ψ(π). nπ Example 3.1. If ψ(s) = s, then a n = b n = 1 n. ψ(s) = s 2, then π n a n π n 3π n b n π n. 4. Conclusion

10 1 M.M. IDDRISU, C.A. OKPOTI, K.A. GBOLAGADE The new Steffensen s inequality (4) is thus proved for continuous functions as well as simple (discontinuous) functions also valid for all functions f L 1 ([,1]). An application of the inequality has also been established for the determination of Fourier coefficients. Conflict of Interests The authors declare that there is no conflict of interests. Acknowledgement The first author wish to thank Frederic SYMESAK for his attention care also deeply thank the French Government for the financial assistance cooperation during his research period at the University of Angers, France. REFERENCES [1] E. F. Beckenbach, Convex Functions, Bull. Amer. Math. Soc. 54 (1948), [2] R. Bellman, On inequalities with alternating signs, Proc. Amer. Math. Soc. 1 (1959), [3] E. K. Godunova V. I. Levin, A general class of inequalities containing Steffensen s inequality, Mat. Zametki 3 (1968), [4] G. H. Hardy, J. E. Littlewood G.Polya, Inequalities, 2nd ed., Cambridge University Press, Cambridge, (1952). [5] D.S. Mitrinovic, J.E. Pecaric, A. M. Fink, Classical New Inequalities in Analysis, Kluwer Academic Publishers, 61, Netherls (1993). [6] D.S. Mitrinovic J.E. Pecaric, On the Bellman Generalization of Steffensen s Inequality III, J. Math. Anal. Appl., 135 (1988), [7] J.E. Pecaric, On the Bellman Generalization of Steffensen s Inequality II, J. Math. Anal. Appl. 14 (1984), [8] J.E. Pecaric, On the Bellman Generalization of Steffensen s Inequality, J. Math. Anal. Appl. 88 (1982), [9] J. Steffensen, Bounds of certain trigonometric integrals, Tenth Scinavian Mathematical congress (1946), , Copenhagan, Hellerup (1947).