Skip to main content

Advertisement

A functional generalization of diamond-α integral Dresher’s inequality on time scales

Article metrics

  • 685 Accesses

Abstract

In this paper, we establish a functional generalization of diamond-α integral Dresher’s inequality on time scales. Its reverse form is also considered.

MSC:26D15, 26E70.

1 Introduction

In the fifties of the previous century, Beckenbach [1] introduced a famous inequality as follows.

Let 1p2 and x i , y i >0, i=1,,n. Then

i = 1 n ( x i + y i ) p i = 1 n ( x i + y i ) p 1 i = 1 n x i p i = 1 n x i p 1 + i = 1 n y i p i = 1 n y i p 1 .
(1.1)

The following integral version of the above-mentioned discrete inequality is due to Dresher [2] (see also [3]):

Assume that f(x) and g(x) are non-negative and continuous real-valued functions on [a,b], and 0<r1p, then

( a b ( f ( x ) + g ( x ) ) p d x a b ( f ( x ) + g ( x ) ) r d x ) 1 / ( p r ) ( a b f p ( x ) d x a b f r ( x ) d x ) 1 / ( p r ) + ( a b g p ( x ) d x a b g r ( x ) d x ) 1 / ( p r ) .
(1.2)

From that time, some generalizations of the Beckenbach-Dresher inequality (1.1) and (1.2) have appeared. Here, we refer to the papers of Pečarić and Beesack [4], Petree and Persson [5], Persson [6] , Varošanec [7], Anwar et al. [8], and Nikolova et al. [9], where the reader can find literature related to this inequality. Recently, Zhao [10] gave the following reverse Dresher’s inequality.

Assume that f(x) and g(x) are non-negative and continuous real-valued functions on [a,b], and p0r1, then

( a b ( f ( x ) + g ( x ) ) p d x a b ( f ( x ) + g ( x ) ) r d x ) 1 / ( p r ) ( a b f p ( x ) d x a b f r ( x ) d x ) 1 / ( p r ) + ( a b g p ( x ) d x a b g r ( x ) d x ) 1 / ( p r ) .
(1.3)

The aim of this work is to give a functional generalization of diamond-α integral Dresher’s inequality for time scales. Its reverse form is also presented.

2 Main results

Let T be a time scale; that is, T is an arbitrary nonempty closed subset of real numbers. The set of the real numbers, the integers, the natural numbers, and the Cantor set are examples of time scales. But the open interval between 0 and 1, the rational numbers, the irrational numbers, and the complex numbers are not time scales. Let a,bT. We now suppose that the reader is familiar with some basic facts from the theory of time scales, which can also be found in [1122], and of delta, nabla and diamond-α dynamic derivatives.

Our main results are given in the following theorems.

Theorem 2.1 (Dresher’s inequality)

Let T be a time scale a,bT with a<b and 0<r1p. Let H l ( x 1 , x 2 ,, x l )>0, F m ( x 1 , x 2 ,, x m ) and G k ( x 1 , x 2 ,, x k ) be three arbitrary functions of l, m and k variables, respectively. Assume that { f i ( x ) } i = 1 m , { g i ( x ) } i = 1 k and { h i ( x ) } i = 1 l are continuous real-valued functions on [ a , b ] T , then

( a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) + G k ( g 1 , g 2 , , g k ) | p α x a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) + G k ( g 1 , g 2 , , g k ) | r α x ) 1 / ( p r ) ( a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) | p α x a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) | r α x ) 1 / ( p r ) + ( a b H l ( h 1 , h 2 , , h l ) | G k ( g 1 , g 2 , , g k ) | p α x a b H l ( h 1 , h 2 , , h l ) | G k ( g 1 , g 2 , , g k ) | r α x ) 1 / ( p r ) ,
(2.1)

there is equality only when the functions | F m ( f 1 , f 2 ,, f m )| and | G k ( g 1 , g 2 ,, g k )| are effectively proportional.

Proof First, we have

( a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) + G k ( g 1 , g 2 , , g k ) | p α x ) 1 / ( p r ) ( ( a b H l | F m | p α x ) 1 / p + ( a b H l | G k | p α x ) 1 / p ) p / ( p r )
(2.2)

by Minkowski’s inequality on time scales [18]. Next, by the right-hand side of the above inequality, we have

( ( a b H l | F m | p α x ) 1 / p + ( a b H l | G k | p α x ) 1 / p ) p / ( p r ) = ( ( a b H l | F m | p α x a b H l | F m | r α x ) 1 / p ( a b H l | F m | r α x ) 1 / p + ( a b H l | G k | p α x a b H l | G k | r α x ) 1 / p ( a b H l | G k | r α x ) 1 / p ) p / ( p r ) .

We apply Hölder’s inequality to the above equality to obtain

( ( a b H l | F m | p α x a b H l | F m | r α x ) 1 / p ( a b H l | F m | r α x ) 1 / p + ( a b H l | G k | p α x a b H l | G k | r α x ) 1 / p ( a b H l | G k | r α x ) 1 / p ) p / ( p r ) ( ( a b H l | F m | p α x a b H l | F m | r α x ) 1 / ( p r ) + ( a b H l | G k | p α x a b H l | G k | r α x ) 1 / ( p r ) ) × ( ( a b H l | F m | r α x ) 1 / r + ( a b H l | G k | r α x ) 1 / r ) r / ( p r ) .
(2.3)

By applying reverse Minkowski’s inequality with 0<r<1, we obtain

( ( a b H l | F m | r α x ) 1 / r + ( a b H l | G k | r α x ) 1 / r ) r a b H l | F m + G k | r α x.
(2.4)

From (2.2), (2.3) and (2.4), we obtain the desired inequality. □

Corollary 2.1 (T=R)

Let 0<r1p. Let H l ( x 1 , x 2 ,, x l )>0, F m ( x 1 , x 2 ,, x m ) and G k ( x 1 , x 2 ,, x k ) be three arbitrary functions of l, m and k variables, respectively. Assume that { f i ( x ) } i = 1 m , { g i ( x ) } i = 1 k and { h i ( x ) } i = 1 l are continuous real-valued functions on [a,b], then

( a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) + G k ( g 1 , g 2 , , g k ) | p d x a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) + G k ( g 1 , g 2 , , g k ) | r d x ) 1 / ( p r ) ( a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) | p d x a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) | r d x ) 1 / ( p r ) + ( a b H l ( h 1 , h 2 , , h l ) | G k ( g 1 , g 2 , , g k ) | p d x a b H l ( h 1 , h 2 , , h l ) | G k ( g 1 , g 2 , , g k ) | r d x ) 1 / ( p r ) ,
(2.5)

there is equality only when the functions | F m ( f 1 , f 2 ,, f m )| and | G k ( g 1 , g 2 ,, g k )| are effectively proportional.

Corollary 2.2 (T=Z)

Let 0<r1p. Let H l ( x 1 , x 2 ,, x l )>0, F m ( x 1 , x 2 ,, x m ) and G k ( x 1 , x 2 ,, x k ) be three arbitrary functions of l, m and k variables, respectively. Assume that { a i 1 , a i 2 , , a i m } i = 1 n , { b i 1 , b i 2 , , b i k } i = 1 n and { c i 1 , c i 2 , , c i l } i = 1 n are real numbers for any m,k,lN, then

( i = 1 n H l ( c i 1 , c i 2 , , c i l ) | F m ( a i 1 , a i 2 , , a i m ) + G k ( b i 1 , b i 2 , , b i k ) | p i = 1 n H l ( c i 1 , c i 2 , , c i l ) | F m ( a i 1 , a i 2 , , a i m ) + G k ( b i 1 , b i 2 , , b i k ) | r ) 1 / ( p r ) ( i = 1 n H l ( c i 1 , c i 2 , , c i l ) | F m ( a i 1 , a i 2 , , a i m ) | p i = 1 n H l ( c i 1 , c i 2 , , c i l ) | F m ( a i 1 , a i 2 , , a i m ) | r ) 1 / ( p r ) + ( i = 1 n H l ( c i 1 , c i 2 , , c i l ) | G k ( b i 1 , b i 2 , , b i k ) | p i = 1 n H l ( c i 1 , c i 2 , , c i l ) | G k ( b i 1 , b i 2 , , b i k ) | r ) 1 / ( p r ) ,
(2.6)

there is equality only when the functions | F m ( a i 1 , a i 2 ,, a i m )| and | G k ( b i 1 , b i 2 ,, b i k )| are effectively proportional.

Theorem 2.2 (reverse Dresher’s inequality)

Let T be a time scale, a,bT with a<b and p0r1. Let H l ( x 1 , x 2 ,, x l )>0, F m ( x 1 , x 2 ,, x m ) and G k ( x 1 , x 2 ,, x k ) be three arbitrary functions of l, m and k variables, respectively. Assume that { f i ( x ) } i = 1 m , { g i ( x ) } i = 1 k and { h i ( x ) } i = 1 l are continuous real-valued functions on [ a , b ] T , then

( a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) + G k ( g 1 , g 2 , , g k ) | p α x a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) + G k ( g 1 , g 2 , , g k ) | r α x ) 1 / ( p r ) ( a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) | p α x a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) | r α x ) 1 / ( p r ) + ( a b H l ( h 1 , h 2 , , h l ) | G k ( g 1 , g 2 , , g k ) | p α x a b H l ( h 1 , h 2 , , h l ) | G k ( g 1 , g 2 , , g k ) | r α x ) 1 / ( p r ) ,
(2.7)

there is equality only when the functions | F m ( f 1 , f 2 ,, f m )| and | G k ( g 1 , g 2 ,, g k )| are effectively proportional.

Proof Let α 1 0, α 2 0, β 1 >0, and β 2 >0, and 1<λ<0, applying the following Radon’s inequality (see [23]):

k = 1 n a k p b k p 1 < ( k = 1 n a k ) p ( k = 1 n b k ) p 1 , x k 0, a k >0,0<p<1,

we have

α 1 λ + 1 β 1 λ + α 2 λ + 1 β 2 λ ( α 1 + α 2 ) λ + 1 ( β 1 + β 2 ) λ ,
(2.8)

there is equality only when (α) and (β) are proportional. Let

α 1 = ( a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) | p α x ) 1 / p ,
(2.9)
β 1 = ( a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) | r α x ) 1 / r ,
(2.10)
α 2 = ( a b H l ( h 1 , h 2 , , h l ) | G k ( g 1 , g 2 , , g k ) | p α x ) 1 / p ,
(2.11)
β 2 = ( a b H l ( h 1 , h 2 , , h l ) | G k ( g 1 , g 2 , , g k ) | r α x ) 1 / r ,
(2.12)

and set λ= r p r . From (2.8)-(2.12), we have

α 1 λ + 1 β 1 λ + α 2 λ + 1 β 2 λ = ( a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) | p α x ) ( λ + 1 ) / p ( a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) | r α x ) λ / r + ( a b H l ( h 1 , h 2 , , h l ) | G k ( g 1 , g 2 , , g k ) | p α x ) ( λ + 1 ) / p ( a b H l ( h 1 , h 2 , , h l ) | G k ( g 1 , g 2 , , g k ) | r α x ) λ / r = ( a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) | p α x a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) | r α x ) 1 / ( p r ) + ( a b H l ( h 1 , h 2 , , h l ) | G k ( g 1 , g 2 , , g k ) | p α x a b H l ( h 1 , h 2 , , h l ) | G k ( g 1 , g 2 , , g k ) | r α x ) 1 / ( p r ) ( α 1 + α 2 ) λ + 1 ( β 1 + β 2 ) λ = [ ( a b H l | F m ( f 1 , , f m ) | p α x ) 1 / p + ( a b H l | G k ( g 1 , , g k ) | p α x ) 1 / p ] p / ( p r ) [ ( a b H l | F m ( f 1 , , f m ) | r α x ) 1 / r + ( a b H l | G k ( g 1 , , g k ) | r α x ) 1 / r ] r / ( p r ) .
(2.13)

Since 1<λ= r p r <0, we may assume p<0<r, and by Minkowski’s inequality for p<0 and 0<r1, we obtain respectively

[ ( a b H l | F m ( f 1 , , f m ) | p α x ) 1 / p + ( a b H l | G k ( g 1 , , g k ) | p α x ) 1 / p ] p a b H l | F m ( f 1 , , f m ) + G k ( g 1 , , g k ) | p α x ,
(2.14)

there is equality only when | F m ( f 1 ,, f m )| and | G k ( g 1 ,, g k )| are proportional, and

[ ( a b H l | F m ( f 1 , , f m ) | r α x ) 1 / r + ( a b H l | G k ( g 1 , , g k ) | r α x ) 1 / r ] r a b H l | F m ( f 1 , , f m ) + G k ( g 1 , , g k ) | r α x
(2.15)

with equality if and only if | F m ( f 1 ,, f m )| and | G k ( g 1 ,, g k )| are proportional.

From equality conditions for (2.8), (2.14) and (2.15), it follows that the sign of equality in (2.7) holds if and only if | F m ( f 1 ,, f m )| and | G k ( g 1 ,, g k )| are proportional.

From (2.13)-(2.15), we arrive at reverse Dresher’s inequality, and the theorem is completely proved. □

Corollary 2.3 (T=R)

Let p0r1. Let H l ( x 1 , x 2 ,, x l )>0, F m ( x 1 , x 2 ,, x m ) and G k ( x 1 , x 2 ,, x k ) be three arbitrary functions of l, m and k variables, respectively. Assume that { f i ( x ) } i = 1 m , { g i ( x ) } i = 1 k and { h i ( x ) } i = 1 l are continuous real-valued functions on [a,b], then

( a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) + G k ( g 1 , g 2 , , g k ) | p d x a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) + G k ( g 1 , g 2 , , g k ) | r d x ) 1 / ( p r ) ( a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) | p d x a b H l ( h 1 , h 2 , , h l ) | F m ( f 1 , f 2 , , f m ) | r d x ) 1 / ( p r ) + ( a b H l ( h 1 , h 2 , , h l ) | G k ( g 1 , g 2 , , g k ) | p d x a b H l ( h 1 , h 2 , , h l ) | G k ( g 1 , g 2 , , g k ) | r d x ) 1 / ( p r ) ,
(2.16)

there is equality only when the functions | F m ( f 1 , f 2 ,, f m )| and | G k ( g 1 , g 2 ,, g k )| are proportional.

Corollary 2.4 (T=Z)

Let p0r1. Let H l ( x 1 , x 2 ,, x l )>0, F m ( x 1 , x 2 ,, x m ) and G k ( x 1 , x 2 ,, x k ) be three arbitrary functions of l, m and k variables, respectively. Assume that { a i 1 , a i 2 , , a i m } i = 1 n , { b i 1 , b i 2 , , b i k } i = 1 n and { c i 1 , c i 2 , , c i l } i = 1 n are real numbers for any m,k,lN, then

( i = 1 n H l ( c i 1 , c i 2 , , c i l ) | F m ( a i 1 , a i 2 , , a i m ) + G k ( b i 1 , b i 2 , , b i k ) | p i = 1 n H l ( c i 1 , c i 2 , , c i l ) | F m ( a i 1 , a i 2 , , a i m ) + G k ( b i 1 , b i 2 , , b i k ) | r ) 1 / ( p r ) ( i = 1 n H l ( c i 1 , c i 2 , , c i l ) | F m ( a i 1 , a i 2 , , a i m ) | p i = 1 n H l ( c i 1 , c i 2 , , c i l ) | F m ( a i 1 , a i 2 , , a i m ) | r ) 1 / ( p r ) + ( i = 1 n H l ( c i 1 , c i 2 , , c i l ) | G k ( b i 1 , b i 2 , , b i k ) | p i = 1 n H l ( c i 1 , c i 2 , , c i l ) | G k ( b i 1 , b i 2 , , b i k ) | r ) 1 / ( p r ) ,
(2.17)

there is equality only when the functions | F m ( a i 1 , a i 2 ,, a i m )| and | G k ( b i 1 , b i 2 ,, b i k )| are proportional.

Obviously, Corollaries 2.2 and 2.4 are well known for the integers.

Remark 2.1 Let { f i ( x , y ) } i = 1 m , { g i ( x , y ) } i = 1 k and { h i ( x , y ) } i = 1 l be continuous real-valued functions on [ a , b ] T × [ a , b ] T , and H l , F m and G k be defined as in Theorem 2.1, then by Theorems 2.1 and 2.2, we obtain functional generalizations of two-dimensional diamond-α integral Dresher’s inequality and reverse Dresher’s inequality on time scales.

References

  1. 1.

    Beckenbach EF: A class of mean value functions. Am. Math. Mon. 1950, 57: 1-6. 10.2307/2305163

  2. 2.

    Dresher M: Moment spaces and inequalities. Duke Math. J. 1953, 20: 261-271. 10.1215/S0012-7094-53-02026-2

  3. 3.

    Beckenbach EF, Bellman R (Eds): Inequalities. Springer, Berlin; 1961.

  4. 4.

    Pečarić JE, Beesack PR: On Jessen’s inequality for convex functions II. J. Math. Anal. Appl. 1986, 118: 125-144. 10.1016/0022-247X(86)90296-9

  5. 5.

    Peetre J, Persson LE: A general Beckenbach’s inequality with applications. Pitman Res. Notes Math. Ser. 211. Function Spaces, Differential Operators and Nonlinear Analysis 1989, 125-139.

  6. 6.

    Persson LE: Generalizations of some classical inequalities with applications. Teubner Texte zur Mathematik 119. Nonlinear Analysis, Function Spaces and Applications Vol. 4 1991, 127-148.

  7. 7.

    Varošanec S: The generalized Beckenbach inequality and related results. Banach J. Math. Anal. 2010, 4(1):13-20. 10.15352/bjma/1272374668

  8. 8.

    Anwar M, Bibi R, Bohner M, Pečarić J: Integral inequalities on time scales via the theory of isotonic linear functionals. Abstr. Appl. Anal. 2011., 2011: Article ID 483595

  9. 9.

    Nikolova L, Persson L-E, Varošanec S: The Beckenbach-Dresher inequality in the Ψ-direct sums of spaces and related results. J. Inequal. Appl. 2012., 2012: Article ID 7

  10. 10.

    Zhao C-J: On Dresher’s inequalities for width-integrals. Appl. Math. Lett. 2012, 25: 190-194. 10.1016/j.aml.2011.08.013

  11. 11.

    Bohner M, Peterson A: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser Boston, Boston; 2001.

  12. 12.

    Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser Boston, Boston; 2003.

  13. 13.

    Ammi MRS, Ferreira RAC, Torres DFM: Diamond- α Jensen’s inequality on time scales. J. Inequal. Appl. 2008., 2008: Article ID 576876 10.1155/2008/576876

  14. 14.

    Malinowska AB, Torres DFM: On the diamond- α Riemann integral and mean value theorems on time scales. Dyn. Syst. Appl. 2009, 18(3-4):469-482.

  15. 15.

    Bohner M, Matthews T, Tuna A: Diamond- α Grüss type inequalities on time scales. Int. J. Dyn. Syst. Differ. Equ. 2011, 3(1/2):234-247.

  16. 16.

    Adamec L: A note on continuous dependence of solutions of dynamic equations on time scales. J. Differ. Equ. Appl. 2011, 17(5):647-656. 10.1080/10236190902873821

  17. 17.

    Adıvar M, Bohner EA: Halanay type inequalities on time scales with applications. Nonlinear Anal., Theory Methods Appl. 2011, 74(18):7519-7531. 10.1016/j.na.2011.08.007

  18. 18.

    Özkan UM, Sarikaya MZ, Yildirim H: Extensions of certain integral inequalities on time scales. Appl. Math. Lett. 2008, 21(10):993-1000. 10.1016/j.aml.2007.06.008

  19. 19.

    Rogers JW Jr., Sheng Q: Notes on the diamond- α dynamic derivative on time scales. J. Math. Anal. Appl. 2007, 326(1):228-241. 10.1016/j.jmaa.2006.03.004

  20. 20.

    Sheng Q: Hybrid approximations via second-order crossed dynamic derivatives with the diamond- α derivative. Nonlinear Anal., Real World Appl. 2008, 9(2):628-640. 10.1016/j.nonrwa.2006.12.006

  21. 21.

    Sheng Q, Fadag M, Henderson J, Davis JM: An exploration of combined dynamic derivatives on time scales and their applications. Nonlinear Anal., Real World Appl. 2006, 7(3):395-413. 10.1016/j.nonrwa.2005.03.008

  22. 22.

    Erbe L: Oscillation criteria for second order linear equations on a time scale. Can. Appl. Math. Q. 2001, 9(4):345-375.

  23. 23.

    Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge; 1934.

Download references

Acknowledgements

The authors thank the editor and the referees for their valuable suggestions to improve the quality of this paper. This paper was partially supported by the Key Laboratory for Mixed and Missing Data Statistics of the Education Department of Guangxi province (No. GXMMSL201404) and the Scientific Research Project of Guangxi Education Department (No. YB2014560).

Author information

Correspondence to Cheng-Dong Wei.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Keywords

  • diamond-α integral
  • time scale
  • Radon’s inequality
  • Dresher’s inequality