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A functional generalization of the reverse Hölder integral inequality on time scales
Mathematical and Computer Modelling 54 (2011) 2939–2942
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Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
A functional generalization of the reverse Hölder integral inequality on time scales Guangsheng Chen a,∗ , Zhan Chen b a
Department of Computer Engineering, Guangxi Modern Vocational Technology College, Hechi, Guangxi, 547000, PR China
b
Academy of Fine Arts, Xuchang University, Xuchang, Henan, 461000, PR China
article
abstract
info
In this paper, we establish a functional generalization of the diamond-α integral reverse Hölder inequality on time scales. Some related inequalities are also considered. © 2011 Elsevier Ltd. All rights reserved.
Article history: Received 26 April 2011 Received in revised form 9 July 2011 Accepted 11 July 2011 Keywords: Diamond-α integral Time scale Hölder inequality Minkowski’s inequality
1. Introduction Let 1/p + 1/q = 1. Assume that f (x) and g (x) are continuous real-valued functions on [a, b]. Then (1) for p > 1, we have the following Hölder inequality (see [1]): b
∫
f (x)g (x)dx ≤
b
∫
a
f (x)dx p
1/p ∫
a
b
g (x)dx q
1/q
.
(1.1)
a
(2) for 0 < p < 1, we have the following reverse Hölder inequality (see [2]): b
∫
f (x)g (x)dx ≥ a
b
∫
1/p ∫
b
f p (x)dx a
1/q
g q (x)dx
.
(1.2)
a
The above inequalities play an important role in many areas of pure and applied mathematics. A large number of generalizations, refinements, variations and applications of (1.1) and (1.2) have been investigated in the literature (see [3–11] and the references therein). Recently, Yang [12] gave a functional generalization of (1.1) on time scales. The aim of this paper is to give a functional generalization of the diamond-α integral reverse Hölder inequality on time scales. Some related inequalities are also considered. Due to restrictions on the number of pages, the basic definitions and theorems of time scale calculus are omitted, and the reader is referred to [13–16].
∗
Corresponding author. E-mail address: [email protected] (G. Chen).
0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.07.015
2940
G. Chen, Z. Chen / Mathematical and Computer Modelling 54 (2011) 2939–2942
2. Main results In this section, we need the following lemma first before we give our results. Lemma 2.1 (See [13]). Let T be a time scale a, b ∈ T with a < b. Assume that f and g are continuous functions on [a, b]T (1) If f (t ) ≥ 0 for all t ∈ [a, b]T , then
b
f (t ) α t ≥ 0. b b f (t ) α t ≤ a g (t ) α t. a b (3) If f (t ) ≥ 0 for all t ∈ [a, b]T , then f (t ) = 0 if and only if a f (t ) α t ≥ 0. a
(2) If f (t ) ≤ g (t ) for all t ∈ [a, b]T , then
By a time scale T , we mean an arbitrary nonempty closed subset of real numbers. The set of real numbers, integers, natural numbers, and the Cantor set are examples of time scales. But rational numbers, irrational numbers, complex numbers, and the open interval between 0 and 1, are not time scales. Theorem 2.2 (The Reverse Hölder Inequality). Let T be a time scale a, b ∈ T with a < b and 0 < p < 1, 1/p + 1/q = 1. Let Hl (x1 , x2 , . . . , xl ) > 0, Fm (x1 , x2 , . . . , xm ) and Gk (x1 , x2 , . . . , xk ) be three arbitrary functions of l, m and k variables, respectively. k l Assume that {fi (x)}m i=1 , {gi (x)}i=1 and {hi (x)}i=1 are continuous real-valued functions on [a, b]T . Then b
∫
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm )Gk (g1 , g2 , . . . , gk )| α x a b
∫
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm )| α x p
≥
1/p
a b
∫
Hl (h1 , h2 , . . . , hl )|Gk (g1 , g2 , . . . , gk )| α x q
1/q
.
(2.1)
a
Moreover, the equality holds if in (2.1) A|Fm (f1 , f2 , . . . , fm )|p = B|Gk (g1 , g2 , . . . , gk )|q , where A and B are constants. Proof. Define Ω (Hl , Fm , Gk ) as the right-hand side of inequality (2.1). If Ω (Hl , Fm , Gk ) = 0, then Fm (f1 , f2 , . . . , fm ) = 0 or Gk (g1 , g2 , . . . , gk ) = 0 and the result follows by item 3 of Lemma 2.1. Then, we assume that Ω (Hl , Fm , Gk ) ̸= 0 and let
φ(x) = b a
Hl (h1 (x), h2 (x), . . . , hl (x))|Fm (f1 (x), f2 (x), . . . , fm (x))|p Hl (h1 (y), h2 (y), . . . , hl (y))|Fm (f1 (y), f2 (y), . . . , fm (y))|p α y
and
ψ(x) = b a
Hl (h1 (x), h2 (x), . . . , hl (x))|Gk (g1 (x), g2 (x), . . . , gk (x))|q Hl (h1 (y), h2 (y), . . . , hl (y))|Gk (g1 (y), g2 (y), . . . , gk (y))|q α y
.
Apply the following reverse Young’s inequality (see [17]) ab ≥
1 p
ap +
1 q
bq ,
a, b > 0, 0 < p < 1, 1/p + 1/q = 1,
with equality holds iff a = b, we have b
∫
φ(x)ψ(x) α x ≥ a
b
∫ a
=
1 p
+
=
1 p
b
a
1
a
q
+
p
+
ψ q (x) q
] α x
Hl (h1 (y), h2 (y), . . . , hl (y))|Fm (f1 (y), f2 (y), . . . , fm (y))|p α y
a
1 q
α x
Hl (h1 (x), h2 (x), . . . , hl (x))|Gk (g1 (x), g2 (x), . . . , gk (x))|q
b
∫
φ p (x)
Hl (h1 (x), h2 (x), . . . , hl (x))|Fm (f1 (x), f2 (x), . . . , fm (x))|p
b
∫
[
b a
Hl (h1 (y), h2 (y), . . . , hl (y))|Gk (g1 (y), g2 (y), . . . , gk (y))|q α y
α x
= 1.
Therefore, we obtain the desired inequality.
Corollary 2.1. (T = R). Let 0 < p < 1 and 1/p+1/q = 1. Let Hl (x1 , x2 , . . . , xl ) > 0, Fm (x1 , x2 , . . . , xm ) and Gk (x1 , x2 , . . . , xk ) k l be three arbitrary functions of l, m and k variables, respectively. Assume that {fi (x)}m i=1 , {gi (x)}i=1 and {hi (x)}i=1 are continuous
G. Chen, Z. Chen / Mathematical and Computer Modelling 54 (2011) 2939–2942
2941
real-valued functions on [a, b]. Then b
∫
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm )Gk (g1 , g2 , . . . , gk )|dx a b
∫
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm )|p dx
≥
1/p ∫
1/q
b
Hl (h1 , h2 , . . . , hl )|Gk (g1 , g2 , . . . , gk )|q dx
.
(2.2)
a
a
Moreover, the equality holds if in (2.2) |Fm (f1 , f2 , . . . , fm )|p = C |Gk (g1 , g2 , . . . , gk )|q , where C is a constant. Corollary 2.2. (T = Z). Let 0 < p < 1 and 1/p + 1/q = 1. Let Hl (x1 , x2 , . . . , xl ) > 0, Fm (x1 , x2 , . . . , xm ) and Gk (x1 , x2 , . . . , xk ) be three arbitrary functions of l, m and k variables, respectively. Assume that {ai1 , ai2 , . . . , aim }ni=1 , {bi1 , bi2 , . . . , bik }ni=1 and {ci1 , ci2 , . . . , cil }ni=1 are real numbers for any m, k, l ∈ N. Then n −
Hl (ci1 , ci2 , . . . , cil )|Fm (ai1 , ai2 , . . . , aim )Gk (bi1 , bi2 , . . . , bk )|
i =1
⩾
n −
1/p Hl (ci1 , ci2 , . . . , cil )|Fm (ai1 , ai2 , . . . , aim )|
p
i=1
n −
1/q Hl (ci1 , ci2 , . . . , cil )|Gk (bi1 , bi2 , . . . , bik )|
q
.
(2.3)
i =1
Moreover, the equality holds if in (2.3) |Fm (ai1 , ai2 , . . . , aim )|p = C |Gk (bi1 , bi2 , . . . , bik )|q , where C is a constant. Theorem 2.3 (Reverse Minkowski’s Inequality). Let T be a time scale a, b ∈ T with a < b and 0 < p < 1. Let Hl (x1 , x2 , . . . , xl ) > 0, Fm (x1 , x2 , . . . , xm ) and Gk (x1 , x2 , . . . , xk ) be three arbitrary functions of l, m and k variables, respectively. k l Assume that {fi (x)}m i=1 , {gi (x)}i=1 and {hi (x)}i=1 are continuous real-valued functions on [a, b]T . Then b
∫
1/p
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm ) + Gk (g1 , g2 , . . . , gk )| α x p
a b
∫
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm )|p α x
≥
1/p
a
1/p
b
∫
Hl (h1 , h2 , . . . , hl )|Gk (g1 , g2 , . . . , gk )|p α x
+
.
(2.4)
a
Moreover, the equality holds if in (2.4) |Fm (f1 , f2 , . . . , fm )|p = C |Gk (g1 , g2 , . . . , gk )|q where C is a constant. Proof. Let
∫
b
∫
a b
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm )|p α x
M =
Hl (h1 , h2 , . . . , hl )|Gk (g1 , g2 , . . . , gk )|p α x
N = a
b
∫
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm )|p α x
W = a ∫
1/p 1/p
b
Hl (h1 , h2 , . . . , hl )|Gk (g1 , g2 , . . . , gk )|p α x
+
.
a
By the Hölder inequality, in view of 0 < p < 1,we have b
∫ W =
∫
a b
≤
Hl (h1 , h2 , . . . , hl )(|Fm (f1 , f2 , . . . , fm )|p M 1/p−1 + |Gk (g1 , g2 , . . . , gk )|p N 1/p−1 ) α x
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm ) + Gk (g1 , g2 , . . . , gk )|p (M 1/p + N 1/p )1−p α x
a
= W 1−p
b
∫
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm ) + Gk (g1 , g2 , . . . , gk )|p α x.
(2.5)
a
By inequality (2.5), we arrive at Minkowski’s inequality and the theorem is completely proved.
2942
G. Chen, Z. Chen / Mathematical and Computer Modelling 54 (2011) 2939–2942
Corollary 2.3. (T = R). Let 0 < p < 1, Hl (x1 , x2 , . . . , xl ) > 0, Fm (x1 , x2 , . . . , xm ) and Gk (x1 , x2 , . . . , xk ) be three arbitrary k l functions of l, m and k variables, respectively. Assume that {fi (x)}m i=1 , {gi (x)}i=1 and {hi (x)}i=1 are continuous real-valued functions on [a, b]. Then b
∫
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm ) + Gk (g1 , g2 , . . . , gk )|p dx
1/p
a b
∫
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm )| dx p
≥
1/p
a b
∫
Hl (h1 , h2 , . . . , hl )|Gk (g1 , g2 , . . . , gk )| dx p
+
1/p
.
(2.6)
a
Moreover, the equality holds if in (2.6) |Fm (f1 , f2 , . . . , fm )|p = C |Gk (g1 , g2 , . . . , gk )|q where C is a constant. Corollary 2.4. (T = Z). Let 0 < p < 1, Hl (x1 , x2 , . . . , xl ) > 0, Fm (x1 , x2 , . . . , xm ) and Gk (x1 , x2 , . . . , xk ) be three arbitrary functions of l, m and k variables, respectively. Assume that {ai1 , ai2 , . . . , aim }ni=1 , {bi1 , bi2 , . . . , bik }ni=1 and {ci1 , ci2 , . . . , cil }ni=1 are real numbers for any m, k, l ∈ N. Then
n −
1/p Hl (ci1 , ci2 , . . . , cil )|Fm (ai1 , ai2 , . . . , aim ) + Gk (bi1 , bi2 , . . . , bk )|
p
i=1
⩾
n −
1/p Hl (ci1 , ci2 , . . . , cil )|Fm (ai1 , ai2 , . . . , aim )|
p
i=1
+
n −
1/p Hl (ci1 , ci2 , . . . , cil )|Gk (bi1 , bi2 , . . . , bik )|
p
.
(2.7)
i=1
Moreover, the equality holds if in (2.7) |Fm (ai1 , ai2 , . . . , aim )|p = C |Gk (bi1 , bi2 , . . . , bik )|q , where C is a constant. Obviously, Corollaries 2.2 and 2.4 are well known for the integers. k l Remark 2.1. Let {fi (x, y)}m i=1 , {gi (x, y)}i=1 and {hi (x, y)}i=1 be continuous real-valued functions on [a, b]T × [a, b]T , and Hl , Fm and Gk be defined as in Theorem 2.2. Then by Theorems 2.2 and 2.3, we obtain functional generalizations of the two dimensional diamond-α integral reverse Hölder inequality and Minkowski’s inequality on time scales.
Acknowledgments The authors thank the referee(s) for useful comments and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
D.S. Mitrinovi’c, Analytic Inequalities, Springer, New York, NY, USA, 1970. J. Kuang, Applied Inequalities, Shandong Science Press, Jinan, 2003. G. Hardy, J.E. Littlewood, G. Pólya, Inequalities, second ed., Cambridge University Press, UK, 1952. X. Yang, A generalization of Holder inequality, J. Math. Anal. Appl. 247 (2000) 328–330. X. Yang, Refinement of Holder inequality and application to Ostrowski inequality, Appl. Math. Comput. 138 (2003) 455–461. X. Yang, A note on Holder inequality, Appl. Math. Comput. 134 (2003) 319–322. X. Yang, Holder’s inequality, Appl. Math. Lett. 16 (2003) 897–903. S. Wu, L. Debnath, Generalizations of Aczél’s inequality and Popoviciu’s inequality, Indian J. Pure Appl. Math. 36 (2) (2005) 49–62. W.S. He, Generalization of a sharp Holder’s inequality and its application, J. Math. Anal. Appl. 332 (2007) 741–750. S. Wu, A new sharpened and generalized version of Holder’s inequality and its applications, Appl. Math. Comput. 197 (2008) 708–714. E.G. Kwon, E.K. Bae, On a continuous form of Holder inequality, J. Math. Anal. Appl. 343 (2008) 585–592. W. Yang, A functional generalization of diamond-α integral Holder’s inequality on time scales, Appl. Math. Lett. 23 (2010) 1208–1212. Moulay R.S. Ammi, Rui A.C. Ferreira, Delfim F.M. Torres, Diamond-α Jensen’s inequality on time scales, J. Inequal. Appl. (2008) 13 pages.. doi:10.1155/2008/576876. Article ID 576876. J.W. Rogers Jr., Q. Sheng, Notes on the diamond-α dynamic derivative on time scales, J. Math. Anal. Appl. 326 (1) (2007) 228–241. Q. Sheng, M. Fadag, J. Henderson, J.M. Davis, An exploration of combined dynamic derivatives on time scales and their applications, Nonlinear Anal. RWA 7 (3) (2006) 395–413. A.B. Malinowska, Delfim F.M. Torres, On the diamond–alpha Riemann integral and mean value theorems on time scales, Dynam. Systems Appl. 18 (3–4) (2009) 469–482. J-S Xing, K-Q Su, P-F tao, The applications of Young inequality and Young inverse inequality, Journal of Zhoukou NormaI University 24 (2) (2007) 37–39 (in chinese).
Contents lists available at SciVerse ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
A functional generalization of the reverse Hölder integral inequality on time scales Guangsheng Chen a,∗ , Zhan Chen b a
Department of Computer Engineering, Guangxi Modern Vocational Technology College, Hechi, Guangxi, 547000, PR China
b
Academy of Fine Arts, Xuchang University, Xuchang, Henan, 461000, PR China
article
abstract
info
In this paper, we establish a functional generalization of the diamond-α integral reverse Hölder inequality on time scales. Some related inequalities are also considered. © 2011 Elsevier Ltd. All rights reserved.
Article history: Received 26 April 2011 Received in revised form 9 July 2011 Accepted 11 July 2011 Keywords: Diamond-α integral Time scale Hölder inequality Minkowski’s inequality
1. Introduction Let 1/p + 1/q = 1. Assume that f (x) and g (x) are continuous real-valued functions on [a, b]. Then (1) for p > 1, we have the following Hölder inequality (see [1]): b
∫
f (x)g (x)dx ≤
b
∫
a
f (x)dx p
1/p ∫
a
b
g (x)dx q
1/q
.
(1.1)
a
(2) for 0 < p < 1, we have the following reverse Hölder inequality (see [2]): b
∫
f (x)g (x)dx ≥ a
b
∫
1/p ∫
b
f p (x)dx a
1/q
g q (x)dx
.
(1.2)
a
The above inequalities play an important role in many areas of pure and applied mathematics. A large number of generalizations, refinements, variations and applications of (1.1) and (1.2) have been investigated in the literature (see [3–11] and the references therein). Recently, Yang [12] gave a functional generalization of (1.1) on time scales. The aim of this paper is to give a functional generalization of the diamond-α integral reverse Hölder inequality on time scales. Some related inequalities are also considered. Due to restrictions on the number of pages, the basic definitions and theorems of time scale calculus are omitted, and the reader is referred to [13–16].
∗
Corresponding author. E-mail address: [email protected] (G. Chen).
0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.07.015
2940
G. Chen, Z. Chen / Mathematical and Computer Modelling 54 (2011) 2939–2942
2. Main results In this section, we need the following lemma first before we give our results. Lemma 2.1 (See [13]). Let T be a time scale a, b ∈ T with a < b. Assume that f and g are continuous functions on [a, b]T (1) If f (t ) ≥ 0 for all t ∈ [a, b]T , then
b
f (t ) α t ≥ 0. b b f (t ) α t ≤ a g (t ) α t. a b (3) If f (t ) ≥ 0 for all t ∈ [a, b]T , then f (t ) = 0 if and only if a f (t ) α t ≥ 0. a
(2) If f (t ) ≤ g (t ) for all t ∈ [a, b]T , then
By a time scale T , we mean an arbitrary nonempty closed subset of real numbers. The set of real numbers, integers, natural numbers, and the Cantor set are examples of time scales. But rational numbers, irrational numbers, complex numbers, and the open interval between 0 and 1, are not time scales. Theorem 2.2 (The Reverse Hölder Inequality). Let T be a time scale a, b ∈ T with a < b and 0 < p < 1, 1/p + 1/q = 1. Let Hl (x1 , x2 , . . . , xl ) > 0, Fm (x1 , x2 , . . . , xm ) and Gk (x1 , x2 , . . . , xk ) be three arbitrary functions of l, m and k variables, respectively. k l Assume that {fi (x)}m i=1 , {gi (x)}i=1 and {hi (x)}i=1 are continuous real-valued functions on [a, b]T . Then b
∫
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm )Gk (g1 , g2 , . . . , gk )| α x a b
∫
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm )| α x p
≥
1/p
a b
∫
Hl (h1 , h2 , . . . , hl )|Gk (g1 , g2 , . . . , gk )| α x q
1/q
.
(2.1)
a
Moreover, the equality holds if in (2.1) A|Fm (f1 , f2 , . . . , fm )|p = B|Gk (g1 , g2 , . . . , gk )|q , where A and B are constants. Proof. Define Ω (Hl , Fm , Gk ) as the right-hand side of inequality (2.1). If Ω (Hl , Fm , Gk ) = 0, then Fm (f1 , f2 , . . . , fm ) = 0 or Gk (g1 , g2 , . . . , gk ) = 0 and the result follows by item 3 of Lemma 2.1. Then, we assume that Ω (Hl , Fm , Gk ) ̸= 0 and let
φ(x) = b a
Hl (h1 (x), h2 (x), . . . , hl (x))|Fm (f1 (x), f2 (x), . . . , fm (x))|p Hl (h1 (y), h2 (y), . . . , hl (y))|Fm (f1 (y), f2 (y), . . . , fm (y))|p α y
and
ψ(x) = b a
Hl (h1 (x), h2 (x), . . . , hl (x))|Gk (g1 (x), g2 (x), . . . , gk (x))|q Hl (h1 (y), h2 (y), . . . , hl (y))|Gk (g1 (y), g2 (y), . . . , gk (y))|q α y
.
Apply the following reverse Young’s inequality (see [17]) ab ≥
1 p
ap +
1 q
bq ,
a, b > 0, 0 < p < 1, 1/p + 1/q = 1,
with equality holds iff a = b, we have b
∫
φ(x)ψ(x) α x ≥ a
b
∫ a
=
1 p
+
=
1 p
b
a
1
a
q
+
p
+
ψ q (x) q
] α x
Hl (h1 (y), h2 (y), . . . , hl (y))|Fm (f1 (y), f2 (y), . . . , fm (y))|p α y
a
1 q
α x
Hl (h1 (x), h2 (x), . . . , hl (x))|Gk (g1 (x), g2 (x), . . . , gk (x))|q
b
∫
φ p (x)
Hl (h1 (x), h2 (x), . . . , hl (x))|Fm (f1 (x), f2 (x), . . . , fm (x))|p
b
∫
[
b a
Hl (h1 (y), h2 (y), . . . , hl (y))|Gk (g1 (y), g2 (y), . . . , gk (y))|q α y
α x
= 1.
Therefore, we obtain the desired inequality.
Corollary 2.1. (T = R). Let 0 < p < 1 and 1/p+1/q = 1. Let Hl (x1 , x2 , . . . , xl ) > 0, Fm (x1 , x2 , . . . , xm ) and Gk (x1 , x2 , . . . , xk ) k l be three arbitrary functions of l, m and k variables, respectively. Assume that {fi (x)}m i=1 , {gi (x)}i=1 and {hi (x)}i=1 are continuous
G. Chen, Z. Chen / Mathematical and Computer Modelling 54 (2011) 2939–2942
2941
real-valued functions on [a, b]. Then b
∫
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm )Gk (g1 , g2 , . . . , gk )|dx a b
∫
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm )|p dx
≥
1/p ∫
1/q
b
Hl (h1 , h2 , . . . , hl )|Gk (g1 , g2 , . . . , gk )|q dx
.
(2.2)
a
a
Moreover, the equality holds if in (2.2) |Fm (f1 , f2 , . . . , fm )|p = C |Gk (g1 , g2 , . . . , gk )|q , where C is a constant. Corollary 2.2. (T = Z). Let 0 < p < 1 and 1/p + 1/q = 1. Let Hl (x1 , x2 , . . . , xl ) > 0, Fm (x1 , x2 , . . . , xm ) and Gk (x1 , x2 , . . . , xk ) be three arbitrary functions of l, m and k variables, respectively. Assume that {ai1 , ai2 , . . . , aim }ni=1 , {bi1 , bi2 , . . . , bik }ni=1 and {ci1 , ci2 , . . . , cil }ni=1 are real numbers for any m, k, l ∈ N. Then n −
Hl (ci1 , ci2 , . . . , cil )|Fm (ai1 , ai2 , . . . , aim )Gk (bi1 , bi2 , . . . , bk )|
i =1
⩾
n −
1/p Hl (ci1 , ci2 , . . . , cil )|Fm (ai1 , ai2 , . . . , aim )|
p
i=1
n −
1/q Hl (ci1 , ci2 , . . . , cil )|Gk (bi1 , bi2 , . . . , bik )|
q
.
(2.3)
i =1
Moreover, the equality holds if in (2.3) |Fm (ai1 , ai2 , . . . , aim )|p = C |Gk (bi1 , bi2 , . . . , bik )|q , where C is a constant. Theorem 2.3 (Reverse Minkowski’s Inequality). Let T be a time scale a, b ∈ T with a < b and 0 < p < 1. Let Hl (x1 , x2 , . . . , xl ) > 0, Fm (x1 , x2 , . . . , xm ) and Gk (x1 , x2 , . . . , xk ) be three arbitrary functions of l, m and k variables, respectively. k l Assume that {fi (x)}m i=1 , {gi (x)}i=1 and {hi (x)}i=1 are continuous real-valued functions on [a, b]T . Then b
∫
1/p
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm ) + Gk (g1 , g2 , . . . , gk )| α x p
a b
∫
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm )|p α x
≥
1/p
a
1/p
b
∫
Hl (h1 , h2 , . . . , hl )|Gk (g1 , g2 , . . . , gk )|p α x
+
.
(2.4)
a
Moreover, the equality holds if in (2.4) |Fm (f1 , f2 , . . . , fm )|p = C |Gk (g1 , g2 , . . . , gk )|q where C is a constant. Proof. Let
∫
b
∫
a b
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm )|p α x
M =
Hl (h1 , h2 , . . . , hl )|Gk (g1 , g2 , . . . , gk )|p α x
N = a
b
∫
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm )|p α x
W = a ∫
1/p 1/p
b
Hl (h1 , h2 , . . . , hl )|Gk (g1 , g2 , . . . , gk )|p α x
+
.
a
By the Hölder inequality, in view of 0 < p < 1,we have b
∫ W =
∫
a b
≤
Hl (h1 , h2 , . . . , hl )(|Fm (f1 , f2 , . . . , fm )|p M 1/p−1 + |Gk (g1 , g2 , . . . , gk )|p N 1/p−1 ) α x
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm ) + Gk (g1 , g2 , . . . , gk )|p (M 1/p + N 1/p )1−p α x
a
= W 1−p
b
∫
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm ) + Gk (g1 , g2 , . . . , gk )|p α x.
(2.5)
a
By inequality (2.5), we arrive at Minkowski’s inequality and the theorem is completely proved.
2942
G. Chen, Z. Chen / Mathematical and Computer Modelling 54 (2011) 2939–2942
Corollary 2.3. (T = R). Let 0 < p < 1, Hl (x1 , x2 , . . . , xl ) > 0, Fm (x1 , x2 , . . . , xm ) and Gk (x1 , x2 , . . . , xk ) be three arbitrary k l functions of l, m and k variables, respectively. Assume that {fi (x)}m i=1 , {gi (x)}i=1 and {hi (x)}i=1 are continuous real-valued functions on [a, b]. Then b
∫
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm ) + Gk (g1 , g2 , . . . , gk )|p dx
1/p
a b
∫
Hl (h1 , h2 , . . . , hl )|Fm (f1 , f2 , . . . , fm )| dx p
≥
1/p
a b
∫
Hl (h1 , h2 , . . . , hl )|Gk (g1 , g2 , . . . , gk )| dx p
+
1/p
.
(2.6)
a
Moreover, the equality holds if in (2.6) |Fm (f1 , f2 , . . . , fm )|p = C |Gk (g1 , g2 , . . . , gk )|q where C is a constant. Corollary 2.4. (T = Z). Let 0 < p < 1, Hl (x1 , x2 , . . . , xl ) > 0, Fm (x1 , x2 , . . . , xm ) and Gk (x1 , x2 , . . . , xk ) be three arbitrary functions of l, m and k variables, respectively. Assume that {ai1 , ai2 , . . . , aim }ni=1 , {bi1 , bi2 , . . . , bik }ni=1 and {ci1 , ci2 , . . . , cil }ni=1 are real numbers for any m, k, l ∈ N. Then
n −
1/p Hl (ci1 , ci2 , . . . , cil )|Fm (ai1 , ai2 , . . . , aim ) + Gk (bi1 , bi2 , . . . , bk )|
p
i=1
⩾
n −
1/p Hl (ci1 , ci2 , . . . , cil )|Fm (ai1 , ai2 , . . . , aim )|
p
i=1
+
n −
1/p Hl (ci1 , ci2 , . . . , cil )|Gk (bi1 , bi2 , . . . , bik )|
p
.
(2.7)
i=1
Moreover, the equality holds if in (2.7) |Fm (ai1 , ai2 , . . . , aim )|p = C |Gk (bi1 , bi2 , . . . , bik )|q , where C is a constant. Obviously, Corollaries 2.2 and 2.4 are well known for the integers. k l Remark 2.1. Let {fi (x, y)}m i=1 , {gi (x, y)}i=1 and {hi (x, y)}i=1 be continuous real-valued functions on [a, b]T × [a, b]T , and Hl , Fm and Gk be defined as in Theorem 2.2. Then by Theorems 2.2 and 2.3, we obtain functional generalizations of the two dimensional diamond-α integral reverse Hölder inequality and Minkowski’s inequality on time scales.
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