A generalized Young inequality and some new results on fractal space

The classical Young inequality [1][2][3][4] is not only interesting in itself but also very useful. The purpose of this work is to establish a generalized Young inequality on fractional set and other inequality based on it. Start with, we review the Yang's fractional set and Yang's geometric expressions for the real line number system.

Theory of the Yang’s fractional set 1.2 Yang’s geometric expressions for the real line number system Geometric representation of real line numbers on a fractional set as points on a real line called the real line axis. For each real line number there correspond one and only one point on the real line [6].

For example, 1 α + 2 α = 3 α . That is, the geometric representation is that cantor set [0,3] is equivalent to the sum of cantor set [0, 1] and cantor set [1,3]. The dimension of cantor set is α , for 0 < α ≤ 1 and, 1 α , 2 α and 3 α are real line numbers on a fractional set. If a α , b α , c α belong to the set R α of real line numbers, then we have the following operation:

(1) a α + b α and a α b α belong to the set R α

(2)

If a α -b α is a nonnegative number, we say that a α is greater than or equal to b α or b α is less than or equal to a α , and write, respectively, a α ≥ b α or b α ≤ a α . If there is no possibility that a α = b α , we write

Geometrically, a α > b α if the point on the real line axis corresponding to a α lies to the left of the point corresponding to b α .

Suppose that a α , b α and c α are any given real line numbers, then we have the following relations:

(1) Either a α > b α , a α = b α or a α < b α ( Law of trichotomy );

(2) If a α > b α and b α > c α , then a α > c α (Law of transitivity);

(3

The formula is similar to classical one in case of α = 1. As direct results, the following inequalities are valid:

In the section, we give the proof of the generalized Young inequality. Here we first start with the generalized Bernoulli’s inequality.

Lemma 2.1 (Generalized Bernoulli’s inequality) Let y > 0

(1) for 0 < m < 1, then

Remark 1. This is classical Bernoulli’s inequality in case of fractal dimension α = 1 [5].

which is equality holding if and only if a p = b q .

Proof. Setting y = a/b, by (2.1) we have

Multiplying both sides by b α in (2.4) gives

Then, we obtain that

Let m = 1 p < 1, then we directly deduce (2.3). Remark 2. (2.3) was proposed [6], here we give its proof.it is classical Young inequality in case of fractal dimension α = 1.

where equality holds if and only if a p = b q .

Proof. For x, y ≥ 0, 0 < p < 1, by (2.3), we have

Hence we complete the proof of Theorem 2.2.

As a direct result, we have the following:

where the equality holds if a

where the equality holds if a p j j = a p k k for ∀j, k .

In the section we discuss some generalizations of holder inequality and Minkowski inequality.

In order to proof our results, we first review the hölder inequality and Minkowski inequality [6]:

Equalities holding if and only if λ

where λ 1 and λ 2 are constants.

Theorem 3.2 .(see [6])

Equalities holding if and only if λ 1 |x i | = λ 2 |y i |, where λ 1 and λ 2 are constants.

Equalities holding if and only if λ

where λ 1 and λ 2 are constants.

Proof. Set c = 1/p, then we have q = -pd, d = c/(c -1). By (3.1), we obtain

In (3.4), multiplying both sides by n i=1

Equalities holding if and only if λ

where λ 1 and λ 2 are constants.

, by Hölder inequality, in view of 0 < p < 1, we have

By inequality (3.7), we arrive at reverse Minkowski’s inequality and the theorem is completely proved.

1/p j = 1.

(1) for p j > 1, we have

Equalities holding if and only if λ j |x ij | = λ k |x ik | for ∀j, k, where λ j and λ k are constants.

(2) for 0 < p 1 < 1, p j < 0, j = 2, . . . m,we have

Equalities holding if and only if λ j |x ij | = λ k |x ik | for ∀j, k, where λ j and λ k are constants. Equalities holding if and only if λ j |x ij | = λ k |x ik | for ∀j, k, where λ j and λ k are constants.

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