Journal of mathematics and computer science 13 (2014), 68-70
A Note on Generalization of Classical Jensenβs Inequality
P.O. Olanipekun, A. Mogbademu
Departmentof Mathematics, Universityof Lagos, Yaba-Nigeria
[email protected], [email protected]
Article history:
Received August 2014
Accepted September 2014
Available online September 2014
Abstract In this note, we prove a new generalisation of the Jensenβs inequality by using a Riemann-Stieltjes
integrable function and convex functions under a mild condition. An example was given to support the claims of this paper.
Keywords: Convex functions, Jensenβs inequality.
1. Introduction and Preliminaries
In [5], Royden and Fitzpatrick, examined the classical form of Jensenβs inequality [3]
π π π₯ ππ₯1
0 β€ (πππ)(π₯)ππ₯
1
0. (1.1)
Using the notion of the supporting line that exists at the point (πΌ, π(πΌ)) for the graph of
π π€βπππ πΌ β 0, 1 . Indeed, they gave a short proof for the Jensenβs inequality. The purpose of
this paper is to employ a simple analytic technique which is independent of the idea in [4] to
show that for any two convex functions π π₯ , π½(π₯) and another Riemann Stieltjes Integrable
function f(x) defined on [a, b] then
π πππ
π β€ π(π)π
π
π (1.2)
under a mild condition.
P.O. Olanipekun, A. Mogbademu / J. Math. Computer Sci. 13 (2014), 68-70
69
Remark: A case where π½(π₯) is the identity function and π β π = 1gives the kind of
Jensenβs inequality discussed in [5].
The following well known definition and Lemmas are useful in the proof of our results.
Definition 1.1. A function π is convex on [a, b] if,
π π₯ β€ π π¦ + π π‘ βπ π¦
π‘βπ¦ π₯ β π¦ ,whereπ β€ π¦ β€ π₯ β€ π‘ β€ π.
Lemma 1.1 ([1, 2][5]). Suppose π is convex on [a, b] and differentiable at πΌ β π, π , then,
π πΌ + πβ² πΌ π₯ β πΌ β€ π π₯ , β π₯ β [a, b] .
Proof: See Lemma 1 of [2] and Theorem 18 in Chapter 6 of [5].
Lemma 1.2 [3]. Let π be an increasing function on the closed bounded interval [a, b], then πβ² is
integrable over [a, b] and πβ² β€ π π β π π .π
π
Proof: See Corollary 4 in section 6. 2 of [5].
2. Main results
Theorem 2.1. Let π π₯ , π½(π₯) be convex functions on (ββ, β) and π(π₯) Riemann-Stieltjes
integrable w.r.t π½(π₯) over [a, b] such that π½ π β π½ π = 1. Then,
π π π₯ ππ
π β€ (πππ)(π₯)π
π
π.
Proof. Let πΌ = πππ
π . (2. 1)
Choose π β β β π¦ = π π‘ β πΌ + π(πΌ) is the equation of the supporting line passing through
(πΌ, π πΌ ) for the graph of π. Clearly, πβ²(πΌβ) < π < πβ²(π+). From Lemma 1. 1, we have:
π π‘ β₯ π π‘ β πΌ + π πΌ β π‘ β β. (2. 2)
And, in particular
π π π₯ β₯ π π π₯ β πΌ + π πΌ πππ π₯ β [π, π] (2. 3)
Integrating both sides of (2.3)
π π π₯ π
π
π
β₯ (π π π₯ β πΌ + π πΌ )π
π
π
= π π π₯ π βππΌ π½ π β π½ π + π πΌ π½ π β π½ π
π
π
P.O. Olanipekun, A. Mogbademu / J. Math. Computer Sci. 13 (2014), 68-70
70
= ππΌ β ππΌ + π πΌ
= π( πππ
π) . (2. 4)
That is, (πππ)ππ
πβ₯ π π π₯ π
π
π completing the Proof.
Example
Let π½ π₯ = 0 , π₯ = π1, π < π₯ β€ π
Clearly, π½ π β π½ π = 1 and for any convex function π and Riemann Integrable function f on [a, b],
then Theorem 2.1 holds.
Acknowlegement: The authors wish to acknowledge the warm support of the Research Group in
Mathematics and Applications of the University of Lagos, Nigeria.
References
[1] H. ASNAASHARI, Some remarks on convexity of Chebysev sets, The Journal of Mathematics and
Computer Science, Vol. 1 No. 2 (2010) 102-106.
[2] H. FINBARR, A note on the Mean Value Theorem, https://cms.math.ca/crux/v22/n7page 290-
293.pdf
[3] R. Kargar, A. Bilavi, S. Abdolahi, S. Maroufi, A class of Multivalent analytic functions defined by a
new linear operator. Journal of Mathematics and Computer Science, 8(2014) 236-334.
[4] D. S. MITRINOVIC, J. E. PECARIC, and A. M. FINK, Classical and New Inequalities in Analysis,
Kluwer Academic Publishers, London, 1993.
[5] H. L. ROYDEN, and P. M. FITZPATRICK, Real Analysis, Fourth Edition, PHI- Learning Private
Limited, New Delhi, 2011.
Top Related