A Hilbert-type fractal integral inequality and its

Liu and Sun Journal of Inequalities and Applications (2017) 2017:83 DOI 10.1186/s13660-017-1360-9

RESEARCH

Open Access

A Hilbert-type fractal integral inequality and its applications Qiong Liu and Wenbing Sun* *

Correspondence: [email protected] Department of Science and Information Science, Shaoyang University, Shaoyang, 422000, P.R. China

Abstract By using thefractal theory and the methods of weight function, a Hilbert-type fractal integral inequality and its equivalent form are given. Their constant factors are proved being the best possible, and their applications are discussed briefly. Keywords: fractal set; Hilbert-type fractal integral inequality; weight function

1 Introduction ∞ ∞ If f , g ≥ , satisfying  <  f  (x) dx < ∞,  <  g  (y) dy < ∞, then there is the following basic Hilbert-type integral inequality and its equivalent form [] ∞  ∞  f (x)g(y) dx dy <  f  (x) dx g  (y) dy , max{x, y}      ∞ ∞ ∞ f (x) f  (x) dx, dx dy <  max{x, y}   

∞ ∞

() ()

where the constants are optimal. Inequalities () and () are important in the analysis and partial differential equations [, ]. In  and , respectively, () and () were generalized and improved by introducing an independent parameter λ and two parameters λ , λ [, ]. In recent years, the fractal theory has been developed rapidly, and it has been widely used in the fields of science and engineering. Some researchers have used the fractal theory to discuss and generalize some classical inequalities on fractal sets [, ], but the research into the Hilbert-type integral inequality on the fractal set is still not involved. In this paper, by using the fractal theory and the method of weight function to make a meaningful attempt, a Hilbert-type integral inequality and its equivalent form on a fractal set are established.

2 Preliminaries Definition . ([]) A non-differentiable function f : R → Rα ( < α ≤ ), x → f (x) is called local fractional continuous at x if for any ε > , there exists δ >  such that |f (x) – f (x )| < εα whenever |x – x | < δ. If f (x) is local fractional continuous on the interval (a, b), we denote f (x) ∈ Cα (a, b). © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided 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.

Liu and Sun Journal of Inequalities and Applications (2017) 2017:83

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Definition . ([]) The local fractional derivative of f (x) of order α ( < α ≤ ) at x is defined by f α (x ) =

dα f (x) Γ (α + )(f (x) – f (x )) = lim , α dx x=x x→x (x – x )α

∞ where Γ (z) =  e–u uz– du (z > ) []. If for all x ∈ I ⊆ R, there exists f (k+)α (x) = k+ α Dx · · · Dαx f (x), then we denote f ∈ D(k+)α (I), where k = , , , . . . . Lemma . ([]) Suppose that f (x) ∈ Cα (a, b) and f (x) ∈ Dα (a, b). Then, for  < α ≤ , we have an α-differential form dα f (x) = f (α) (x) dxα . Lemma . ([]) Let I be an interval, f , g : I ⊆ R → Rα (I  is the interior of I) such that f , g ∈ Dα (I  ). Then the following differentiation rules are valid: α (i) d (f (x)±g(x)) = f (α) (x) ± g (α) (x); dxα (ii)

dα (f (x)g(x)) dxα f (x)

(iii) (iv)

dα g(x)

=

dxα dα (Cf (x)) dxα

= f (α) (x)g(x) + f (x)g (α) (x);

f (α) (x)g(x)–f (x)g (α) (x) g  (x)

= Cf

(α)

(g(x) = );

(x), where C is a constant;

(v) If y(x) = (f g)(x), then

dα y(x) dxα

= f (α) (g(x))(g () (x))α .

Definition . ([]) Let f (x) ∈ Cα (a, b). Then the local fractional integral is defined by α a Ib f (x) =

 Γ (α + )

 f (ti )(ti )α , lim Γ (α + ) λ→ i= N

b

f (t)(dt)α = a

with ti = ti – ti– (i = , . . . , N ) and λ = max≤i≤N {ti }, and a = t < t < · · · < tN = b is partition of interval [a, b]. Here, it follows that a Ibα =  if a = b, a Ibα f (x) = –b Iaα f (x) if a < b. Lemma . ([]) () Suppose that f (x) = g (α) (x) ∈ Cα (a, b), then we have α a Ib f (x) = g(b) – g(a);

() Suppose that f (x), g(x) ∈ Dα (a, b), and f (α) (x), g (α) (x) ∈ Cα (a, b), then we have α (α) b a Ib f (x)g (x) = f (x)g(x)|a

– a Ibα f (α) (x)g(x).

Lemma . ([]) For f (x) = xγ (γ > ), we have the following equations: Γ ( + γ ) γ –α dα (xγ ) x ; = α dx Γ ( + γ – α) b Γ ( + γ ) γ +α  b xγ (dx)α = – aγ +α . Γ (α + ) a Γ ( + γ + α)


Page 3 of 8

Lemma . ([, ]) If f , g (≥ ) ∈ Cα (a, b), F, G, h (≥ ) ∈ Cα (S(β) ), p > , a fractal surface, then we have (i) Hölder’s inequality on the fractal set

 p

+

 q

= , S(β) is

b  f (x)g(x)(dx)α Γ (α + ) a p q b b   p α q α ≤ f (x)(dx) g (x)(dx) ; Γ (α + ) a Γ (α + ) a (ii) Hölder’s weighted inequality on the fractal set  h(x, y)F(x, y)G(x, y)(dx)α (dy)α Γ  (α + ) S(β) p  p α α ≤ h(x, y)F (x, y)(dx) (dy) Γ  (α + ) S(β) q  p α α × h(x, y)G (x, y)(dx) (dy) . Γ  (α + ) S(β) The inequality keeps the form of equality, then there exist constants A and B such that they are not all zero and AF p (x, y) = BGq (x, y) a.e. on S(β) . Lemma . Suppose that p > ,  ω(α, p, x) := Γ (α + ) ω(α, q, y) :=

 Γ (α + )

+

 q

= ,  < α ≤ , and weight functions are defined by α

∞

 y–  α pα (dy) , max{xα , yα } x– q

∞

x–   α qα (dx) , max{xα , yα } y– p



 p



x ∈ (, +∞),

α

y ∈ (, +∞).

Then α

α

ω(α, p, x) = η(α)x  (p–) ,

ω(α, q, y) = η(α)y  (q–) ,

where η(α) =

α+ . Γ ( + α)

()

Proof Set xy = u, then (dy)α = xα (du)α . Note the following exchange integral, let u = t , and √ α u = s, we have (du)α = –t –α (dt)α and u–  (du)α = α (ds)α . Then we have

∞

α

y–   α pα (dy) max{xα , yα } x– q  ∞  max{, uα } – α α u  (du)α = x  (p–) Γ (α + )  max{xα , yα }  ∞  α α α = x  (p–) u–  (du)α + u–  (du)α Γ (α + )  

 ω(α, p, x) = Γ (α + )


Page 4 of 8

   α α u–  (du)α + t –  (dt)α Γ (α + )     α α = x  (p–) u–  (du)α Γ (α + )   α+ α (p–) α  (ds) =x Γ (α + )  α

= x  (p–)

=

α+ α x  (p–) Γ ( + α) α

= η(α)x  (p–) . α

Similarly, we obtain ω(α, q, y) = η(α)y  (q–) .

Lemma . Suppose that p > , p + q = ,  < α ≤ , and ε >  is small enough, let us define the real functions as follows: f (x) =

, x ∈ (, ), – α – αε p , x ∈ [, ∞), x

g(y) =

, y ∈ (, ), – α – αε q , y ∈ [, ∞), x

then we have J · εα =

α  I∞

p

α

x  (p–) f (x)

α h · ε α =  I∞

α  I∞

p

α  I∞

α

y  (q–) g q (y)

q

· εα =

η(α) f (x)g(y) · εα >  – o() α α max{x , y } Γ (α + )

 , Γ (α + )

()

ε → + .

()

Proof Note the properties of local fractal space [, ]: (a + b)α = aα + bα and (–a)α = –aα , we easily obtain J · εα =

α  I∞

α (p–) p p α α (q–) q q α x f (x) g (y) ·ε  I∞ y 

α –α(+ε) p α –α(+ε) q α x =  I∞ ·ε =  I∞ y 

ε



ε

α

 . Γ (α + )

αε

Let t  – q = u, x–  + q = v, and from (  – qε )α t –  – q (dt)α = (du)α , –(  – qε )α x– (dv)α , we write –α α – α – αε α · I  t  q =  I∞ x x

  Γ (α + )

∞

 ( 

=– =

–

ε α  ) Γ (α q

α

αε

t –  – q (dt)α





 =  ε α  (  – q ) Γ (α + ) =

 x

x–α (dx)α

α + αε q 

∞

–α

x (dx)

α



+ )

α + αε q 



 (  – qε )α Γ  (α + )

 . (  – qε )α Γ  (α + )

x–



(dv)α 

(dx)α

ε

(dv)α 

∞



– + x  q

(dx)α =


Further, let

y x

Page 5 of 8

= t, and by Lemma ., we have

f (x)g(y) · εα max{xα , yα } – α – αεq – α – αε α y α =  I∞ x  p  I∞ · εα max{xα , yα } – α – αεq α –α(+ε) t α · εα =  I∞ x I x ∞ max{, t α } α –α(+ε) x =  I∞ – α – αεq – α – αεq – α – αεq t t t α α α +  I∞ – I  · εα ×  I α α x max{, t } max{, t } max{, t α } – α – αεq – α – αε α –α(+ε) α – α – αε t α α q q  +  I∞ t  – I  · εα =  I∞ x  I t x max{, t α } α αε α αε α –α(+ε) α – α – αε  q x +  Iα t –  + q –  I α t –  – q · εα =  I∞  I t

α h · ε α =  I∞

α  I∞

x

=

 ( 

–

ε α  ) Γ (α q

+ )

+

 ( 

+

ε α  ) Γ (α q

+ )

α αε α –  I∞ x–α(+ε) ·  I α t –  – q · εα x

=

–α(+ε) α – α – αε α η(α) α x · I  t  q · ε + o () –  I∞ x Γ (α + )

>

–α α – α – αε α η(α) α + o () –  I∞ x · I  t  q · ε x Γ (α + )

=

η(α) εα + o () –  ε α  Γ (α + ) (  – q ) Γ (α + )

=

η(α)  – o() Γ (α + )

ε → + .

3 Main results and applications α α Introducing the mark:  I∞ [ I∞ F(x, y)] =

 Γ  (α+)

∞∞ 



F(x, y)(dx)α (dy)α (see []). α

α Theorem . If p > , p + q = ,  < α ≤ , f , g (> ) ∈ Cα (, ∞), and  <  I∞ (x  (p–) f p (x)) < α α (y  (q–) g q (y)) < ∞, then ∞,  <  I∞

α α  I∞  I∞

α α (p–) p p α α (q–) q q f (x)g(y) x < η(α)  I∞ f (x) g (y) ,  I∞ y  α α max{x , y }

()

where the constant factor η(α) defined in () is the best possible. Proof By Hölder’s weighted inequality on the fractal set and Lemma ., we obtain α  I∞

α  I∞

∞ ∞  f (x)g(y) f (x)g(y) = (dx)α (dy)α α , yα } max{xα , yα } Γ  (α + )  max{x  – pα – qα ∞ ∞  x f (x)g(y) y =  (dx)α (dy)α α α α – –α Γ (α + )  max{x , y } x q  y p

 ≤  Γ (α + )



∞ ∞ 

α

f (x)g(y) y–  α α pα (dx) (dy) max{xα , yα } x– q

p


 × Γ  (α + )

Page 6 of 8

∞ ∞





α

f (x)g(y) x–  α α qα (dx) (dy) max{xα , yα } y– p

q

p ∞  p α ω(α, p, x)f (x)(dx) Γ  (α + )  q ∞ ∞  q α × ω(α, q, y)g (y)(dy) Γ  (α + )   p ∞  α = η(α) x  (p–) f p (x)(dx)α Γ (α + )  q ∞  α (q–) q α  × y g (y)(dy) Γ (α + )  α α (p–) p p α α (q–) q q x = η(α)  I∞ f (x) g (y) .  I∞ y 

=

()

Now assume that equality holds in (), there exist two nonzero constants A and B such –α 

–α

that A y– pα f p (x) = B x– qα a.e. in (, ∞) × (, ∞), then there is constant C =  such that x q α (p–) p 

y p α (q–) q 

Ax f (x) = By g (y) = C a.e. in (, ∞) × (, ∞). Assuming that A = , we have ∞ C α  C α α ∞ x  (p–) f p (x) = CA a.e. in (, ∞). Because Γ (α+)  A (dx) = AΓ (α+) x | is diffuse, which α α (x  (p–) f p (x)) < ∞, thus inequality () is strict. contradicts the fact that  <  I∞ If the constant factor η(α) in () is not optimal, then there exists positive K < η(α) such that inequality () is still valid if we replace η(α) by K . Hence by () and (), we have η(α)( – o()) < K . Letting ε → + , we get K ≥ η(α), which contradicts the fact that K < η(α), therefore η(α) in () is the best possible. Theorem . Under the conditions of Theorem ., we have α  I∞

α(–q) α (q–) y  I∞

f (x) max{xα , yα }

p

α (p–) p α < ηp (α) I∞ f (x) , x

()

where the constant factor ηp (α) is the best possible, and inequality () is equivalent to inequality (). α

α (x  (p–) f p (x)) < ∞, there exists n ∈ N Proof Define [f (x)]n := min{n, f (x)}. Since  <  I∞ α

α(–q)

p

p

[f (x)]n  q such that  <  Inα (x  (p–) [f (x)]n ) < ∞ (n ≥ n ). Setting gn (y) := y (q–) [  Inα max{x α ,yα } ] ( n < n n y < n, n ≥ n ), when n ≥ n , by (), we find

α  <  Inα y  (q–) gnq (y) n n  α = y  (q–) gnq– (y)gn (y)(dy)α Γ (α + ) n  = Γ (α + ) =

 Γ  (α + )

< η(α)

n α I n n

 n

n n

 n

 n

pq α(–q) [f (x)]n [f (x)] n α y (q–)  In (dy)α n max{xα , yα } max{xα , yα }

[f (x)]n gn (y) (dx)α (dy)α max{xα , yα }

α p  α  I α x  (p–) f (x) n p  Inα y  (q–) gnq (y) q .

 n n

n

()


Page 7 of 8

Moreover, by () we have 
) ∈ C. (, ∞),  <  I∞ (f (x)) < ∞,  <  I∞ (g (y)) < ∞, then we have the following equivalence inequalities: f (x)g(y)  .   .   ) ∈ C. (, ∞),  <  I∞ ∞, then we have the following equivalence inequalities:

.   .   f (x)g(y) f (x) , < η(.)  I∞  I∞ g (y) max{x. , y. }  f (x) . . .  f (x) , < η (.) I∞  I∞  I∞ max{x. , yn } . .  I∞  I∞

() ()

where the constant factors η(.), η (.) are the best values.

5 Conclusions In the paper, based on the local fractional calculus theory, a Hilbert-type fractional integral inequality and its equivalent form are tentatively researched. The results show that some methods and skills of the Hilbert-type integral inequality can be transplanted to the research of Hilbert-type fractional integral inequality, which provides a new direction and field to research Hardy-Hilbert’s integral inequalities. Competing interests The authors declare that they have no competing interests. Authors’ contributions The two authors contributed equally to this work. They all read and approved the final version of the manuscript. Acknowledgements The authors are extremely grateful to the reviewers for a critical reading of the manuscript and making valuable comments and suggestions leading to an overall improvement of the paper. This work was supported by the National Natural Science Foundations of China (No. 11171280) and Scientific Support Project of Hunan Province Education Department of China (No. 10C1186).

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