1. IntroductionIn the following, we present a result proved by Mitrinović and Pečarić as given in [1] and ( [2] , p. 235).
Theorem 1. Let g i ∈ G ( f i , k ) for ( i = 1 , 2 ) be a class, where f i ( x ) for ( i = 1 , 2 ) are continuous functions and f 2 ( x ) > 0 implies g 2 ( x ) > 0 for every x ∈ [ a , b ] and g i : [ a , b ] → ℝ are represented by
g i ( x ) : = ∫ a b k ( x , y ) f i ( y ) d y , ∀ x ∈ [ a , b ] , i = 1 , 2 ,
where k ( x , y ) is nonnegative arbitrary kernel. Consider w ( x ) ≥ 0 for every x ∈ [ a , b ] . Let F : ℝ 0 + = [ 0 , ∞ ) → ℝ be a convex and increasing function, then the following inequality holds
∫ a b w ( x ) F ( | g 1 ( x ) g 2 ( x ) | ) d x ≤ ∫ a b F ( | f 1 ( y ) f 2 ( y ) | ) s ( y ) d y , (1)
where,
s ( y ) : = f 2 ( y ) ∫ a b w ( x ) k ( x , y ) g 2 ( x ) d x , ∀ y ∈ [ a , b ] , g 2 ( x ) ≠ 0.
Next we present a result on diamond-α calculus, as given in [3] .
Theorem 2. Let T 1 , T 2 be two time scales, and a , b ∈ T 1 ; c , d ∈ T 2 ; k ( x , y ) is a kernel function with x ∈ [ a , b ] T 1 , y ∈ [ c , d ] T 2 ; k is continuous function from [ a , b ] T 1 × [ c , d ] T 2 into ℝ 0 + = [ 0 , ∞ ) . Consider
K ( x ) : = ∫ c d k ( x , y ) ⋄ α y , ∀ x ∈ [ a , b ] T 1 .
We assume that K ( x ) > 0 , ∀ x ∈ [ a , b ] T 1 . Consider f : [ c , d ] T 2 → ℝ continuous, and the ⋄ α -integral operator function
g ( x ) : = ∫ c d k ( x , y ) f ( y ) ⋄ α y , ∀ x ∈ [ a , b ] T 1 .
Consider also the weight function w : [ a , b ] T 1 → ℝ 0 + , which is continuous.
Define further the function s ( y ) : = ∫ a b w ( x ) k ( x , y ) K ( x ) ⋄ α x , ∀ y ∈ [ c , d ] T 2 . Let I denote
any of ( 0, ∞ ) or [ 0, ∞ ) , and F : I → ℝ be a convex and increasing function. In particular, we assume that
| f | ( [ c , d ] T 2 ) ⊆ I .
Then
∫ a b w ( x ) F ( | g ( x ) | K ( x ) ) ⋄ α x ≤ ∫ c d s ( y ) F ( | f ( y ) | ) ⋄ α y . (2)
We extend these results on time scale calculus. In this paper, it is assumed that all considerable integrals exist and are finite and T is a time scale, a , b ∈ T with a < b and an interval [ a , b ] T means the intersection of a real interval with the given time scale.
2. PreliminariesWe need here basic concepts of delta calculus. The results of delta calculus are adapted from [4] [5] [6] .
Time scale calculus was initiated by Stefan Hilger as given in [7] . A time scale is an arbitrary nonempty closed subset of the real numbers. It is denoted by T . For t ∈ T , forward jump operator σ : T → T is defined by
σ ( t ) : = inf { s ∈ T : s > t } .
The mapping μ : T → ℝ 0 + = [ 0 , ∞ ) such that μ ( t ) : = σ ( t ) − t is called the forward graininess function. The backward jump operator ρ : T → T is defined by
ρ ( t ) : = sup { s ∈ T : s < t } .
The mapping ν : T → ℝ 0 + = [ 0 , ∞ ) such that ν ( t ) : = t − ρ ( t ) is called the backward graininess function. If σ ( t ) > t , we say that t is right-scattered, while if ρ ( t ) < t , we say that t is left-scattered. Also, if t < sup T and σ ( t ) = t , then t is called right-dense, and if t > inf T and ρ ( t ) = t , then t is called left-dense. Points that are right-dense and left-dense at the same time are called dense. If T has a left-scattered maximum M, then T k = T − { M } . Otherwise T k = T .
For a function f : T → ℝ , the derivative f Δ is defined as follows. Let t ∈ T k , if there exists f Δ ( t ) ∈ ℝ such that for all ϵ > 0 , there exists a neighborhood U of t, such that
| f ( σ ( t ) ) − f ( s ) − f Δ ( t ) ( σ ( t ) − s ) | ≤ ϵ | σ ( t ) − s | ,
for all s ∈ U , then f is said to be delta differentiable at t, and f Δ ( t ) is called the delta derivative of f at t.
A function f : T → ℝ is said to be right-dense continuous (rd-continuous), if it is continuous at each right-dense point and there exists a finite left limit in every left-dense point. The set of all rd-continuous functions is denoted by C r d ( T , ℝ ) .
The next definition is given in [4] [5] [6] .
Definition 1. A function F : T → ℝ is called a delta antiderivative of f : T → ℝ , provided that F Δ ( t ) = f ( t ) holds for all t ∈ T k , then the delta integral of f is defined by
∫ a b f ( t ) Δ t = F ( b ) − F ( a ) .
The following results of nabla calculus are taken from [4] [5] [6] [8] .
If T has a right-scattered minimum m, then T k = T − { m } . Otherwise T k = T . The function f : T → ℝ is called nabla differentiable at t ∈ T k , if there exists f ∇ ( t ) ∈ ℝ such that for any ϵ > 0 , there exists a neighborhood V of t, such that
| f ( ρ ( t ) ) − f ( s ) − f ∇ ( t ) ( ρ ( t ) − s ) | ≤ ϵ | ρ ( t ) − s | ,
for all s ∈ V .
A function f : T → ℝ is left-dense continuous (ld-continuous), provided it is continuous at left-dense points in T and its right-sided limits exist (finite) at right-dense points in T . The set of all ld-continuous functions is denoted by C l d ( T , ℝ ) .
The next definition is given in [4] [5] [6] [8] .
Definition 2. A function G : T → ℝ is called a nabla antiderivative of g : T → ℝ , provided that G ∇ ( t ) = g ( t ) holds for all t ∈ T k , then the nabla integral of g is defined by
∫ a b g ( t ) ∇ t = G ( b ) − G ( a ) .
Now we present short introduction of diamond-α derivative as given in [4] [9] .
Let T be a time scale and f ( t ) be differentiable on T in the Δ and ∇ senses. For t ∈ T k k , where T k k = T k ∩ T k , diamond-α dynamic derivative f ⋄ α ( t ) is defined by
f ⋄ α ( t ) = α f Δ ( t ) + ( 1 − α ) f ∇ ( t ) , 0 ≤ α ≤ 1.
Thus f is diamond-α differentiable if and only if f is Δ and ∇ differentiable.
The diamond-α derivative reduces to the standard Δ -derivative for α = 1 , or the standard ∇ -derivative for α = 0 . It represents a weighted dynamic derivative for α ∈ ( 0,1 ) .
Theorem 3. [9] : Let f , g : T → ℝ be diamond-α differentiable at t ∈ T . Then
1) f ± g : T → ℝ is diamond-α differentiable at t ∈ T , with
( f ± g ) ⋄ α ( t ) = f ⋄ α ( t ) ± g ⋄ α ( t ) .
2) f g : T → ℝ is diamond-α differentiable at t ∈ T , with
( f g ) ⋄ α ( t ) = f ⋄ α ( t ) g ( t ) + α f σ ( t ) g Δ ( t ) + ( 1 − α ) f ρ ( t ) g ∇ ( t ) .
3) For g ( t ) g σ ( t ) g ρ ( t ) ≠ 0 , f g : T → ℝ is diamond-α differentiable at
t ∈ T , with
( f g ) ⋄ α ( t ) = f ⋄ α ( t ) g σ ( t ) g ρ ( t ) − α f σ ( t ) g ρ ( t ) g Δ ( t ) − ( 1 − α ) f ρ ( t ) g σ ( t ) g ∇ ( t ) g ( t ) g σ ( t ) g ρ ( t ) .
Theorem 4. [9] : Let a , t ∈ T and h : T → ℝ . Then the diamond-α integral from a to t of h is defined by
∫ a t h ( s ) ⋄ α s = α ∫ a t h ( s ) Δ s + ( 1 − α ) ∫ a t h ( s ) ∇ s , 0 ≤ α ≤ 1,
provided that there exist delta and nabla integrals of h on T .
Theorem 5. [9] : Let a , b , t ∈ T , c ∈ ℝ . Assume that f ( s ) and g ( s ) are ⋄ α -integrable functions on [ a , b ] T , then
1) ∫ a t [ f ( s ) ± g ( s ) ] ⋄ α s = ∫ a t f ( s ) ⋄ α s ± ∫ a t g ( s ) ⋄ α s ;
2) ∫ a t c f ( s ) ⋄ α s = c ∫ a t f ( s ) ⋄ α s ;
3) ∫ a t f ( s ) ⋄ α s = − ∫ t a f ( s ) ⋄ α s ;
4) ∫ a t f ( s ) ⋄ α s = ∫ a b f ( s ) ⋄ α s + ∫ b t f ( s ) ⋄ α s ;
5) ∫ a a f ( s ) ⋄ α s = 0 .
We need the following results.
Theorem 6. [4] : Let a , b ∈ T and c , d ∈ ℝ . Suppose that g ∈ C ( [ a , b ] T , ( c , d ) ) and h ∈ C ( [ a , b ] T , ℝ ) with ∫ a b | h ( s ) | ⋄ α s > 0 . If F ∈ C ( ( c , d ) , ℝ ) is convex, then generalized Jensen’s inequality is
F ( ∫ a b | h ( s ) | g ( s ) ⋄ α s ∫ a b | h ( s ) | ⋄ α s ) ≤ ∫ a b | h ( s ) | F ( g ( s ) ) ⋄ α s ∫ a b | h ( s ) | ⋄ α s . (3)
If F is strictly convex, then the inequality ≤ can be replaced by < .
Theorem 7. [3] [10] : Let a , b ∈ T . Let f i ∈ C ( [ a , b ] T , ℝ ) , i = 1 , ⋯ , n are ⋄ α -
integrable functions and p i > 1 such that ∑ i = 1 n 1 p i = 1 . Then
∫ a b ∏ i = 1 n | f i ( t ) | ⋄ α t ≤ ∏ i = 1 n ( ∫ a b | f i ( t ) | p i ⋄ α t ) 1 p i , (4)
which is generalized Rogers-Hölder’s Inequality.
Definition 3. [11] : A function f : T → ℝ is called convex on I T = I ∩ T , where I is an interval of ℝ (open or closed), if
f ( λ t + ( 1 − λ ) s ) ≤ λ f ( t ) + ( 1 − λ ) f ( s ) , (5)
for all t , s ∈ I T and all λ ∈ [ 0,1 ] such that λ t + ( 1 − λ ) s ∈ I T .
The function f is strictly convex on I T if (5) is strict for distinct t , s ∈ I T and λ ∈ ( 0,1 ) .
The function f is concave (respectively, strictly concave) on I T , if − f is convex (respectively, strictly convex).
3. Main ResultsFirst we present ⋄ α -integral general fractional Schlömilch’s type inequalities on time scales, which is an extension of Schlömilch’s inequality given in [12] .
Theorem 8. Let [ a , b ] T 1 and [ c , d ] T 2 be two time scales;
k ( x , y ) : [ a , b ] T 1 × [ c , d ] T 2 → ℝ 0 + is continuous kernel function with x ∈ [ a , b ] T 1 and y ∈ [ c , d ] T 2 . Let ⋄ α -integral operator functions g i : [ a , b ] T 1 → ℝ belonging to a class G ( f i , k ) for ( i = 1 , 2 ) are represented by
g i ( x ) : = ∫ c d k ( x , y ) f i ( y ) ⋄ α y ,
where f i : [ c , d ] T 2 → ℝ are continuous functions. Continuous weight function is defined by w : [ a , b ] T 1 → ℝ 0 + with ∫ a b w ( x ) ⋄ α x = 1 . Define
s ( y ) : = f 2 ( y ) ∫ a b w ( x ) k ( x , y ) g 2 ( x ) ⋄ α x and ∀ y ∈ [ c , d ] T 2 , where f 2 ( y ) > 0 implies
g 2 ( x ) > 0 . Let F : ℝ 0 + = [ 0, ∞ ) → ℝ 0 + be a convex and increasing function.
If η 2 ≥ η 1 ≥ 1 , then the following inequality holds
( ∫ a b w ( x ) F ( | g 1 ( x ) g 2 ( x ) | ) η 1 ⋄ α x ) 1 η 1 ≤ ( ∫ c d F ( | f 1 ( y ) f 2 ( y ) | ) η 2 s ( y ) ⋄ α y ) 1 η 2 . (6)
Proof. In order to prove this Theorem, we need Bernoulli’s inequality, that is, if x > 0 , then
p x + 1 − p ≤ x p , if p ≥ 1.
Since η 2 ≥ η 1 ≥ 1 , we have η 2 η 1 ≥ 1 . Thus, by Bernoulli’s inequality, we have
∫ a b w ( x ) ( F ( | g 1 ( x ) g 2 ( x ) | ) ∫ a b w ( x ) F ( | g 1 ( x ) g 2 ( x ) | ) ⋄ α x ) η 2 η 1 ⋄ α x ≥ ∫ a b w ( x ) ( η 2 η 1 F ( | g 1 ( x ) g 2 ( x ) | ) ∫ a b w ( x ) F ( | g 1 ( x ) g 2 ( x ) | ) ⋄ α x + 1 − η 2 η 1 ) ⋄ α x = 1 ,
that is,
∫ a b w ( x ) F ( | g 1 ( x ) g 2 ( x ) | ) η 2 η 1 ⋄ α x ≥ ( ∫ a b w ( x ) F ( | g 1 ( x ) g 2 ( x ) | ) ⋄ α x ) η 2 η 1 .
Let F be replaced by F η 1 and taking power 1 η 2 > 0 , we get
( ∫ a b w ( x ) F ( | g 1 ( x ) g 2 ( x ) | ) η 1 ⋄ α x ) 1 η 1 ≤ ( ∫ a b w ( x ) F ( | g 1 ( x ) g 2 ( x ) | ) η 2 ⋄ α x ) 1 η 2 = ( ∫ a b w ( x ) F ( | ∫ c d k ( x , y ) f 1 ( y ) ⋄ α y g 2 ( x ) | ) η 2 ⋄ α x ) 1 η 2 = ( ∫ a b w ( x ) F ( | 1 g 2 ( x ) ∫ c d k ( x , y ) f 2 ( y ) f 1 ( y ) f 2 ( y ) ⋄ α y | ) η 2 ⋄ α x ) 1 η 2
≤ ( ∫ a b w ( x ) ( 1 g 2 ( x ) ∫ c d k ( x , y ) f 2 ( y ) F ( | f 1 ( y ) f 2 ( y ) | ) η 2 ⋄ α y ) ⋄ α x ) 1 η 2 = ( ∫ c d F ( | f 1 ( y ) f 2 ( y ) | ) η 2 ( f 2 ( y ) ∫ a b w ( x ) k ( x , y ) g 2 ( x ) ⋄ α x ) ⋄ α y ) 1 η 2 = ( ∫ c d F ( | f 1 ( y ) f 2 ( y ) | ) η 2 s ( y ) ⋄ α y ) 1 η 2 ,
where we used the generalized Jensen’s inequality and Fubini’s theorem.
This proves the claim. □
Remark. If we set η 1 = η 2 = 1 and F : [ 0, ∞ ) → ℝ be a convex and increasing function, then (6) takes the form
∫ a b w ( x ) F ( | g 1 ( x ) g 2 ( x ) | ) ⋄ α x ≤ ∫ c d F ( | f 1 ( y ) f 2 ( y ) | ) s ( y ) ⋄ α y . (7)
If [ a , b ] T 1 = [ c , d ] T 2 , where T 1 = T 2 = ℝ , then (7) takes the form of (1).
Corollary 1. If η 1 = η 2 = 1 , F : [ 0, ∞ ) → ℝ be a convex and increasing function and α = 1 , then delta version form of (6) is
∫ a b w ( x ) F ( | g 1 ( x ) g 2 ( x ) | ) Δ x ≤ ∫ c d F ( | f 1 ( y ) f 2 ( y ) | ) s ( y ) Δ y . (8)
If η 1 = η 2 = 1 , F : [ 0, ∞ ) → ℝ be a convex and increasing function and α = 0 , then nabla version form of (6) is
∫ a b w ( x ) F ( | g 1 ( x ) g 2 ( x ) | ) ∇ x ≤ ∫ c d F ( | f 1 ( y ) f 2 ( y ) | ) s ( y ) ∇ y . (9)
Remark. Now we take that F is not necessarily increasing and is taken from
( 0, ∞ ) into ℝ 0 + and f 1 ( y ) f 2 ( y ) has fixed and strict sign, then according to
Theorem 8, we get
( ∫ a b w ( x ) F ( | g 1 ( x ) g 2 ( x ) | ) η 1 ⋄ α x ) 1 η 1 ≤ ( ∫ c d F ( | f 1 ( y ) f 2 ( y ) | ) η 2 s ( y ) ⋄ α y ) 1 η 2 .
Corollary 2. If we apply for F ( x ) = x p , p > 1 , then (6) takes the form
( ∫ a b w ( x ) ( | g 1 ( x ) g 2 ( x ) | ) p η 1 ⋄ α x ) 1 η 1 ≤ ( ∫ c d ( | f 1 ( y ) f 2 ( y ) | ) p η 2 s ( y ) ⋄ α y ) 1 η 2 . (10)
Corollary 3. If we apply for F ( x ) = e x , x ≥ 0 , then (6) takes the form
( ∫ a b w ( x ) e η 1 ( | g 1 ( x ) g 2 ( x ) | ) ⋄ α x ) 1 η 1 ≤ ( ∫ c d e η 2 ( | f 1 ( y ) f 2 ( y ) | ) s ( y ) ⋄ α y ) 1 η 2 . (11)
Corollary 4. If η 1 = η 2 = 1 , F : ( 0, ∞ ) → ℝ be a convex and not necessarily
increasing function, f 1 ( y ) f 2 ( y ) has fixed and strict sign and we apply for
F ( x ) = − ln x , x > 0 , then (6) takes the form
∫ a b w ( x ) ln ( | g 1 ( x ) g 2 ( x ) | ) ⋄ α x ≥ ∫ c d ln ( | f 1 ( y ) f 2 ( y ) | ) s ( y ) ⋄ α y . (12)
Remark. If we set f 2 ( y ) = 1 , g 1 ( x ) = g ( x ) , f 1 ( y ) = f ( y ) , η 1 = η 2 = 1 and F : [ 0, ∞ ) → ℝ be a convex and increasing function, then
g 2 ( x ) = ∫ c d k ( x , y ) ⋄ α y = K ( x ) , ∀ x ∈ [ a , b ] T 1 .
We assume that K ( x ) > 0 , and define
s ( y ) : = ∫ a b w ( x ) k ( x , y ) K ( x ) ⋄ α x , ∀ y ∈ [ c , d ] T 2 .
Then (6) takes the form of (2), as proved in [3] .
Corollary 5. If we take T 1 = q ℕ 0 , q > 1 , where ℕ 0 is the set of nonnegative integers and T 2 = ℝ .
Then
∫ q m q n f ( x ) ⋄ α x = ( q − 1 ) ∑ i = m n − 1 q i [ α f ( q i ) + ( 1 − α ) f ( q i + 1 ) ] ,
for [ a , b ] T 1 = [ q m , q n ] q ℕ 0 , m < n , where m , n ∈ ℕ 0 .
And
∫ c d f ( y ) ⋄ α y = ∫ c d f ( y ) d y .
When η 1 = η 2 = 1 and F : [ 0, ∞ ) → ℝ be a convex and increasing function, then (6) can be written as
( q − 1 ) ∑ i = m n − 1 q i [ α w ( q i ) F ( | g 1 ( q i ) g 2 ( q i ) | ) + ( 1 − α ) w ( q i + 1 ) F ( | g 1 ( q i + 1 ) g 2 ( q i + 1 ) | ) ] ≤ ∫ c d F ( | f 1 ( y ) f 2 ( y ) | ) s ( y ) d y .
We can generalize Theorem 8 for convex functions of several variables on time scales in the upcoming theorem.
Theorem 9. Let [ a , b ] T 1 and [ c , d ] T 2 be two time scales;
k ( x , y ) : [ a , b ] T 1 × [ c , d ] T 2 → ℝ 0 + is continuous kernel function with x ∈ [ a , b ] T 1 and y ∈ [ c , d ] T 2 . Let ⋄ α -integral operator functions g i : [ a , b ] T 1 → ℝ belonging to a class G ( f i , k ) for ( i = 1 , 2 , 3 ) are represented by
g i ( x ) : = ∫ c d k ( x , y ) f i ( y ) ⋄ α y ,
where f i : [ c , d ] T 2 → ℝ are continuous functions. Continuous weight function is defined by w : [ a , b ] T 1 → ℝ 0 + with ∫ a b w ( x ) ⋄ α x = 1 . Define
s ( y ) : = f 2 ( y ) ∫ a b w ( x ) k ( x , y ) g 2 ( x ) ⋄ α x and ∀ y ∈ [ c , d ] T 2 , where f 2 ( y ) > 0 implies
g 2 ( x ) > 0 . Let F : ℝ 0 + × ℝ 0 + = [ 0, ∞ ) × [ 0, ∞ ) → ℝ 0 + be a convex and increasing function.
If η 2 ≥ η 1 ≥ 1 , then the following inequality holds
( ∫ a b w ( x ) F ( | g 1 ( x ) g 2 ( x ) | , | g 3 ( x ) g 2 ( x ) | ) η 1 ⋄ α x ) 1 η 1 ≤ ( ∫ c d F ( | f 1 ( y ) f 2 ( y ) | , | f 3 ( y ) f 2 ( y ) | ) η 2 s ( y ) ⋄ α y ) 1 η 2 . (13)
Proof. Proof is similar to Theorem 8. □
Remark. If we set η 1 = η 2 = 1 , F : [ 0, ∞ ) × [ 0, ∞ ) → ℝ be a convex and increasing function and [ a , b ] T 1 = [ c , d ] T 2 , where T 1 = T 2 = ℝ , then (13) reduces to
∫ a b w ( x ) F ( | g 1 ( x ) g 2 ( x ) | , | g 3 ( x ) g 2 ( x ) | ) d x ≤ ∫ a b F ( | f 1 ( y ) f 2 ( y ) | , | f 3 ( y ) f 2 ( y ) | ) s ( y ) d y ,
as given in ( [2] , p. 236).
Now we present ⋄ α -integral general fractional Rogers-Holder’s type inequalities.
Upcoming result is an application of general fractional Schlömilch’s type dynamic inequality.
Theorem 10. Let [ a , b ] T 1 and [ c , d ] T 2 be two time scales; k i ( x , y ) : [ a , b ] T 1 × [ c , d ] T 2 → ℝ 0 + for i = 1 , ⋯ , n ∈ ℕ are continuous kernel functions with x ∈ [ a , b ] T 1 and y ∈ [ c , d ] T 2 . Let ⋄ α -integral operator functions f i , g i : [ a , b ] T 1 → ℝ for i = 1 , ⋯ , n ∈ ℕ are represented by
f i ( x ) : = ∫ c d k i ( x , y ) u i ( y ) ⋄ α y ,
and
g i ( x ) : = ∫ c d k i ( x , y ) v i ( y ) ⋄ α y ,
where u i , v i : [ c , d ] T 2 → ℝ are continuous functions for i = 1 , ⋯ , n ∈ ℕ . Continuous weight function is defined by w : [ a , b ] T 1 → ℝ 0 + with
∫ a b w ( x ) ⋄ α x = 1 . Define s i ( y ) : = v i ( y ) ∫ a b w ( x ) k i ( x , y ) g i ( x ) ⋄ α x , and ∀ y ∈ [ c , d ] T 2 for
i = 1 , ⋯ , n ∈ ℕ , where v i ( y ) > 0 implies g i ( x ) > 0 for i = 1 , ⋯ , n ∈ ℕ . Let F i : ℝ 0 + → ℝ 0 + for i = 1 , ⋯ , n ∈ ℕ are convex and increasing functions.
If p i > 1 with ∑ i = 1 n 1 p i < 1 . Then the following inequality holds
∫ a b w ( x ) ∏ i = 1 n F i ( | f i ( x ) g i ( x ) | ) ⋄ α x ≤ ∏ i = 1 n ( ∫ c d F i ( | u i ( y ) v i ( y ) | ) p i s i ( y ) ⋄ α y ) 1 p i . (14)
Proof. Let γ : = ∑ i = 1 n 1 p i < 1 and ζ i : = γ p i < p i for i = 1 , ⋯ , n . Then ∑ i = 1 n 1 ζ i = 1 ,
where ζ i > 1 for i = 1 , ⋯ , n . We use here generalized Rogers-Hölder’s inequality, Schlömilch’s inequality, generalized Jensen’s inequality and Fubini’s theorem, as
∫ a b w ( x ) ∏ i = 1 n F i ( | f i ( x ) g i ( x ) | ) ⋄ α x = ∫ a b ∏ i = 1 n ( w ( x ) 1 ζ i F i ( | f i ( x ) g i ( x ) | ) ) ⋄ α x ≤ ∏ i = 1 n ( ∫ a b w ( x ) F i ( | f i ( x ) g i ( x ) | ) ζ i ⋄ α x ) 1 ζ i
≤ ∏ i = 1 n ( ∫ a b w ( x ) F i ( | f i ( x ) g i ( x ) | ) p i ⋄ α x ) 1 p i = ∏ i = 1 n ( ∫ a b w ( x ) F i ( | ∫ c d k i ( x , y ) u i ( y ) ⋄ α y g i ( x ) | ) p i ⋄ α x ) 1 p i
= ∏ i = 1 n ( ∫ a b w ( x ) F i ( | 1 g i ( x ) ∫ c d k i ( x , y ) v i ( y ) u i ( y ) v i ( y ) ⋄ α y | ) p i ⋄ α x ) 1 p i ≤ ∏ i = 1 n ( ∫ a b w ( x ) ( 1 g i ( x ) ∫ c d k i ( x , y ) v i ( y ) F i ( | u i ( y ) v i ( y ) | ) p i ⋄ α y ) ⋄ α x ) 1 p i = ∏ i = 1 n ( ∫ c d F i ( | u i ( y ) v i ( y ) | ) p i ( v i ( y ) ∫ a b w ( x ) k i ( x , y ) g i ( x ) ⋄ α x ) ⋄ α y ) 1 p i = ∏ i = 1 n ( ∫ c d F i ( | u i ( y ) v i ( y ) | ) p i s i ( y ) ⋄ α y ) 1 p i .
This proves the claim. □
Corollary 6. If we apply for F i ( x ) = x ξ i , x ≥ 0 , i = 1 , ⋯ , n and let ξ i ≥ 1 , i = 1 , ⋯ , n . Then (14) takes the form
∫ a b w ( x ) ∏ i = 1 n ( | f i ( x ) g i ( x ) | ) ξ i ⋄ α x ≤ ∏ i = 1 n ( ∫ c d ( | u i ( y ) v i ( y ) | ) ξ i p i s i ( y ) ⋄ α y ) 1 p i . (15)
.