Some Estimations Čebyšev-Grüss Type Inequalities Involving Functions and their Derivatives

Author(s): Mehmet Zeki Sarikaya1, Sümeyra Kaplan1
1Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce-Turkey.
Copyright © Mehmet Zeki Sarikaya, Sümeyra Kaplan. 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, some inequalities related to Čebyšev’s functional are proved.

Keywords: Čebyšev inequalities, Grüss inequalities.

1. Introduction

The classical form of Grüss inequality, first published by G. Grüss in 1935, gives an estimate of the difference between the integral of the product and the product of the integrals of two functions. In recent years, several bounds for the Cebysev functional in various cases including convexity assumptions for the functions involved are proved. In the subsequent years, many variants of these inequalities appeared in the literature (see, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]). In 1935, G. Grüss [4] proved the following inequality:
|1baabf(x)g(x)dx1baabf(x)dx1baabg(x)dx|14(Φφ)(Γγ),
(1)
provided that f and g are two integrable function on [a,b] satisfying the condition φf(x)Φ and γg(x)Γ for all x[a,b]. The constant 14 is best possible. In 1882, P. L. Čebyšev [13] gave the following inequality: |T(f,g)|112(ba)2fg, where f,g:[a,b]R are absolutely continuous function, whose first derivatives f and g are bounded, T(f,g)=1baabf(x)g(x)dx(1baabf(x)dx)(1baabg(x)dx) and . denotes the norm in L[a,b] defined as p=esssupt[a,b]|p(t)|.
In [14], Beesack et al. have proved the following Cebysev inequality for absolutely continuous functions whose first derivatives belong to Lp spaces.
|T(f,g)|(ba)4(2p1p(p+1))1p(2q1q(q+1))1qfpgq
(2)
where hp=(ab|h(x)|pdx)1p, p>1 and 1p+1q=1.
In this paper, some inequalities related to Chebyshev’s functional are proved. We give our results in the case of differentiable functions whose derivatives and theirself belong to Lp[a,b], 1p.

2. Main results

Theorem 2.1. Let f,g:[a,b]R be an absolutely continuous function on [a,b] so that |f| and |g| are convex on [a,b].

  1. If f,f,g,gL[a,b], then we have
    |T(f,g)|(ba)6[gf+fg],
    (3)
  2. If f,f,g,gLp[a,b], p>1, 1p+1q=1, then we have
    |T(f,g)|21q2(ba)2q1[(q+1)(q+2)]1q[(ba)1p(gfp+fgp)+gpfp+fpgp],
    (4)
  3. If f,f,g,gL1[a,b], then we have
    |T(f,g)|14[gf1+fg1]+14(ba)[g1f1+f1g1].
    (5)

Proof. For any x,t[a,b], xt, we write f(x)f(t)xt=1xttxf(u)du=01f[(1λ)x+λt]dλ and so

f(x)=f(t)+(xt)01f[(1λ)x+λt]dλ.
(6)
Let’s rewrite (6) as follows
g(x)=g(t)+(xt)01g[(1λ)x+λt]dλ.
(7)
Multiplying (6) by g(x) and (7) by f(x), adding the resulting identities, and integrate over x,t[a,b], and divide by (ba)2, we have 2(ba)abf(x)g(x)dx=2(ba)2abf(x)dxabg(x)dx+1(ba)2abab(xt)g(x)01f[(1λ)x+λt]dλdtdx+1(ba)2abab(xt)f(x)01g[(1λ)x+λt]dλdtdx and rewriting we get
T(f,g)=12(ba)2abab(xt)g(x)01f[(1λ)x+λt]dλdtdx+12(ba)2abab(xt)f(x)01g[(1λ)x+λt]dλdtdx.
(8)
(1) Thus, using the properties of modulus and the convexity of |f| and |g|, we have |T(f,g)|12(ba)2abab|xt||g(x)|01[(1λ)|f(x)|+λ|f(t)|]dλdtdx+12(ba)2abab|xt||f(x)|01[(1λ)|g(x)|+λ|g(t)|]dλdtdx=14(ba)2abab|xt|[|g(x)||f(x)|+|g(x)||f(t)|]dtdx+14(ba)2abab|xt|[|f(x)||g(x)|+|f(x)||g(t)|]dtdx14(ba)2esssupx[a,b][|g(x)||f(x)|+|f(x)||g(x)|]abab|xt|dxdt+14(ba)2esssupx[a,b]|g(x)|abab|xt||f(t)|dxdt+14(ba)2esssupx[a,b]|f(x)|abab|xt||g(t)|dxdt14(ba)2[gf+fg]ab[(ta)2+(bt)22]dt+14(ba)2gesssupt[a,b]|f(t)|ab[(ta)2+(bt)22]dt+14(ba)2fesssupt[a,b]|g(t)|ab[(ta)2+(bt)22]dt=(ba)6[gf+fg] for x,t[a,b], and the inequality (3) is proved.
(2) As above, we rewrite
|T(f,g)|14(ba)2abab|xt|[|g(x)||f(x)|+|f(x)||g(x)|]dtdx+14(ba)2abab|xt|[|g(x)||f(t)|+|f(x)||g(t)|]dtdx.
(9)
Using the Hölder’s inequality for p>1, 1p+1q=1, we have |T(f,g)|14(ba)2(abab|xt|qdtdx)1q×{(abab[|g(x)||f(x)|+|f(x)||g(x)|]pdtdx)1p+(abab|g(x)|p|f(t)|pdtdx)1p+(abab|f(x)|p|g(t)|pdtdx)1p}=14(ba)2[(ba)1p|gf|+|fg|p+gpfp+fpgp]×(ab[(ta)q+1+(bt)q+1q+1]dt)1q21q2(ba)2q1[(q+1)(q+2)]1q[(ba)1p(gfp+fgp)+gpfp+fpgp] and the inequality (4) is proved.
(3) We consider the inequality (9) that |T(f,g)|14(ba)ab[|g(x)||f(x)|+|f(x)||g(x)|]supt[a,b]|xt|dx+14(ba)2absupt[a,b]|xt|ab[|g(x)||f(t)|+|f(x)||g(t)|]dtdx=14(ba)ab[|g(x)||f(x)|+|f(x)||g(x)|]max{xa,bx}dx+14(ba)2abmax{xa,bx}[|g(x)|f1+|f(x)|g1]dx=14(ba)ab[|g(x)||f(x)|+|f(x)||g(x)|]((ba)+|2xba|2)dx+14(ba)2ab((ba)+|2xba|2)[|g(x)|f1+|f(x)|g1]dx18[gf1+fg1]+18(ba)[g1f1+f1g1]+18(ba)supx[a,b]|2xba|{ab[|g(x)||f(x)|+|f(x)||g(x)|]dx+1(ba)ab[|g(x)|f1+|f(x)|g1]dx}=14[gf1+fg1]+14(ba)[g1f1+f1g1] and the inequality (5) is proved.

Theorem 2.2. Let f,g:[a,b]R be an absolutely continuous function on [a,b] so that |f|p and |g|p with p>1are convex on [a,b].

  1. If f,f,g,gL[a,b], then we have
    |T(f,g)|(ba)6[gf+fg],
    (10)
  2. If f,f,g,gLp[a,b], p>1, 1p+1q=1, then we have
    |T(f,g)|21q1p1(ba)2q1[(q+1)(q+2)]1q×{((ba)gfpp+gppfpp)1p+((ba)fgpp+fppgpp)1p},
    (11)
  3. If f,f,g,gLp[a,b], then we have
    |T(f,g)|(ba)1q2{[fpp+gfpp]1p+[gpp+fgpp]1p}.
    (12)

Proof. By Hölder’s inequality and using the convexity of |f|p, we get 01f[(1λ)x+λt]dλ(011q)1q(01|f[(1λ)x+λt]|pdλ)1p=(01|f[(1λ)x+λt]|pdλ)1p(01[(1λ)|f(x)|p+λ|f(t)|p]dλ)1p=(|f(x)|p+|f(t)|p2)1p. From (8) and using the properties of modulus and the convexity of |f|p and |g|p, we have |T(f,g)|12(ba)2abab|xt||g(x)|(|f(x)|p+|f(t)|p2)1pdtdx+12(ba)2abab|xt||f(x)|(|g(x)|p+|g(t)|p2)1pdtdx for x,t[a,b].
(1) If we take f,f,g,gL[a,b], then from ( 13), we have |T(f,g)|12(ba)2esssupx[a,b]|g(x)|esssupx,t[a,b](|f(x)|p+|f(t)|p2)1pabab|xt|dtdx+12(ba)2esssupx[a,b]|f(x)|esssupx,t[a,b](|g(x)|p+|g(t)|p2)1pabab|xt|dtdx(ba)6[gf+fg] for any x,t[a,b], the inequality is proved.
(2) If f,f,g,gLp[a,b], p>1, 1p+1q=1, then from (13) and by Hölder’s inequality we have |T(f,g)|12(ba)2(abab|xt|qdtdx)1q×{(abab|g(x)|p(|f(x)|p+|f(t)|p2)dtdx)1p+(abab|f(x)|p(|g(x)|p+|g(t)|p2)dtdx)1p}=121+1p(ba)2(ab(xa)q+1+(bx)q+1q+1dx)1q×((ba)gfpp+gppfpp)1p+((ba)fgpp+fppgpp)1p=21q1p1(ba)2q1[(q+1)(q+2)]1q[((ba)gfpp+gppfpp)1p+((ba)fgpp+fppgpp)1p] which is proved the inequality (11).
(3) If f,f,g,gLp[a,b], then from (13) and by Hölder’s inequality we also have |T(f,g)|12(ba)2absupx[a,b]|xt|ab|g(x)|(|f(x)|p+|f(t)|p2)1pdxdt+12(ba)2absupx[a,b]|xt|ab|f(x)|(|g(x)|p+|g(t)|p2)1pdxdt=121+1p(ba)2abmax{ta,bt}ab|g(x)|(|f(x)|p+|f(t)|p)1pdxdt+121+1p(ba)2abmax{ta,bt}ab|f(x)|(|g(x)|p+|g(t)|p)1pdxdt12(ba)2ab((ba)+|2tba|2)×[(ab|g(x)|p(|f(x)|p+|f(t)|p)dx)1p(ab1qdx)1q]dt+12(ba)2ab((ba)+|2tba|2)×[(ab|f(x)|p(|g(x)|p+|g(t)|p)dx)1p(ab1qdx)1q]dt (ba)1q22supt[a,b]((ba)+|2tba|2)ab[(ba)|f(t)|p+gfpp]1pdt+(ba)1q22supt[a,b]((ba)+|2tba|2)ab[(ba)|g(t)|p+fgpp]1pdt(ba)1q12(ab[(ba)|f(t)|p+gfpp]dt)1p(ab1qdt)1q+(ba)1q12(ab[(ba)|g(t)|p+fgpp]dt)1p(ab1qdt)1q=(ba)1q2{[fpp+gfpp]1p+[gpp+fgpp]1p} which is proved the inequality (12).

Competing interests

The authors declare that they have no competing interests.

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