Fractional Quantum Integral Inequalities
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Research Article Fractional Quantum Integral Inequalities ¨ g˘ unmez ¨ Hasan O ¨ and Umut Mutlu Ozkan Department of Mathematics, Faculty of Science and Arts, Kocatepe University, 03200 Afyonkarahisar, Turkey ¨ Correspondence should be addressed to Umut Mutlu Ozkan, umut [email protected] Received 10 November 2010; Revised 19 January 2011; Accepted 16 February 2011 Academic Editor: J. Szabados ¨ g˘ unmez ¨ Copyright q 2011 H. O and U. M. Ozkan. 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. The aim of the present paper is to establish some fractional q-integral inequalities on the specific time scale, Ìt0 {t : t t0 qn , n a nonnegative integer} ∪ {0}, where t0 ∈ Ê, and 0 < q < 1.
1. Introduction The study of fractional q-calculus in 1 serves as a bridge between the fractional qcalculus in the literature and the fractional q-calculus on a time scale Ìt0 {t : t t0 qn , n a nonnegative integer} ∪ {0}, where t0 ∈ Ê, and 0 < q < 1. Belarbi and Dahmani 2 gave the following integral inequality, using the RiemannLiouville fractional integral: if f and g are two synchronous functions on 0, ∞, then Γα 1 α J α fg t ≥ J ftJ α gt, tα
1.1
for all t > 0, α > 0. Moreover, the authors 2 proved a generalized form of 1.1, namely that if f and g are two synchronous functions on 0, ∞, then tα tβ J β fg t J α fg t ≥ J α ftJ β gt J β ftJ α gt, Γα 1 Γ β1 for all t > 0, α > 0, and β > 0.
1.2
2
Journal of Inequalities and Applications
Furthermore, the authors 2 pointed out that if fi i1,2,...,n are n positive increasing functions on 0, ∞, then n n 1−n fi t ≥ J α f1 J α fi t, J α
i1
1.3
i1
for any t > 0, α > 0. In this paper, we have obtained fractional q-integral inequalities, which are quantum versions of inequalities 1.1, 1.2, and 1.3, on the specific time scale Ìt0 {t : t t0 qn , n a nonnegative integer} ∪ {0}, where t0 ∈ Ê, and 0 < q < 1. In general, a time scale is an arbitrary nonempty closed subset of the real numbers 3. Many authors have studied the fractional integral inequalities and applications. For example, we refer the reader to 4–6. To the best of our knowledge, this paper is the first one that focuses on fractional qintegral inequalities.
2. Description of Fractional q-Calculus Let t0 ∈ Ê and define
Ìt
0
t : t t0 qn , n a nonnegative integer ∪ {0},
0 < q < 1.
If there is no confusion concerning t0 , we will denote Ìt0 by Ì. For a function f : nabla q-derivative of f is f qt − ft ∇q ft q−1 t
2.1
Ì
→
Ê, the
2.2
for all t ∈ Ì \ {0}. The q-integral of f is t
∞
fs∇s 1 − q t qi f tqi .
2.3
i0
0
The fundamental theorem of calculus applies to the q-derivative and q-integral; in particular, ∇q
t
fs∇s ft,
2.4
0
and if f is continuous at 0, then t 0
∇q fs∇s ft − f0.
2
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