The Quotient Inequalities

This paper contributes to the field of inequalities, specifically, the relationships among the norms of products of elements or vectors or functions and their quotients. Thus, we established that the norm of product of two vectors or functions is less than or equal to the norm of its quotient if the norm of the denominator is less than or equal to one. On the other hand, we prove that the norm of the quotient of two vectors or functions is less than or equal to the norm of their product. In addition, we introduce the proofs of inequalities including the norm of index power of products and their quotients, and then applied these inequalities to estabish properties of some functional spaces, as well as, extention of some of the results in these functional spaces. 2010 Mathematics Subject Classifications: 44B56, 44B57


Introduction
Inequalities play a central role in mathematical analysis with numerous applications in solving ill-posed differential equations, approximation theory, optimization theory, numerical analysis, probability theory and statistics. The authors in [1] obtained estimates relating martingale difference sequences in the complex uniformly convex spaces. In [2], they obtained some estimates for geometric inequalities and compared these inequalities. The authors in [3] provided an alternative way of proving the quasi-normed linear space through binomial inequalities.
In this paper, the new inequalities; first and second quotient inequalities, for the continuous mappings or operators on a real Hilbert space, complex Hilbert space, Banach space and Hölder's spaces and Sobolev spaces are provided. These quotient inequalities are used in obtaining various new inequalities for continuous functions of both selfadjoint and adjoint operators.
The section one of this paper contains the general overview of the inequalities with emphasizes on the quotient inequalities. In section 2, the preliminary results including

Some Preliminary Results
In this section, the definitions regarding the quotient inequalities and index power inequalities are provided.
Definition 1 (First and Second Product Inequalities). Let a 1 and a 2 be any two positive real numbers, then See [4].
Definition 3 (Young's Inequality). For 1 < p < ∞, q the conjugate of p, and any two positive numbers a and b, then Definition 4 (Contractive Mapping). Let T : X → X be a mapping from a complete normed linear space X into itself. The Lipschitz continuity on T is said to be a contraction if See [7].
Definition 5. Let γ ∈ (0, 1]. We say a function T : X → Y is Hölder continuous of where L is boundedness constant may depend on X, x o , γ and T . See [8]. Theorem 1 (Gagliardo-Nirenberg-Sobolev inequality). Let 1 ≤ p < n. Then there exists a constant C > 0 (depending on p and n) such that In particular, we have the continuous imbedding For example, see authors in [9].
then we have the Ostrowski type inequality for selfadjoint operators: Moreover, we have See [10].

Main Result
In this section, we derive both the first and second quotient inequalities in a suitable functional space.
Theorem 2 (First Quotient Inequality). Suppose that x and y are two real numbers or two real-valued vectors, then where the equality occurs at either x = 0 or y = 1.
Proof : Setting f (x, y) = −νx 2 − 2νx 2 y 2 , the function attains its maximum value at zero, for all (x, y) ∈ R 2 and ν ∈ [0, 1]. We can see that: Setting ν = 1 in inequality (1) yields We search for the regions for which x and y hold. Setting ν = 1 4 in inequality (1), we obtain Again, plugging ν = 1 2 into inequality (1) yields From the inequalities (3) and (4), we consider three situations for which inequality (2) holds. We observe that Also, we can see that the two inequalities in (3) and (4) are equal if, Lastly, the norms on the left hand sides of inequalities (3) and (4) can be written as: The region for which y holds is as follows.
Combining the inequalities (5), (7) and (8), and equation (6) together with inequality (2), we obtain Theorem 3 (Second Quotient Inequality ). Suppose that x and y are any two real numbers or any two real-valued vectors, then where the equality occurs at either x = y = 1 or y = 1.

Illustration of the First Quotient Inequality to the Real Line
In this subsection, the illustration of the first quotient inequality is provided. and Ω |g(x)|dx ≤ 1, Proof : We observe that: Applying the first quotient inequality to the term on the right hand side of the above inequality, we obtain

The Applications of the Second Quotient Inequality to L P Spaces
In this subsection, the second quotient inequality is used to estimate the integrals in L p spaces. and Ω |g(x)|dx ≥ 1, Proof : We can see that: Applying the second quotient inequality to the term on the right hand side of the above inequality yields Corollary 1. Suppose that f (x) and g(x) are any two measurable functions defined on R with f (x) ≤ 1 and g(x) Corollary 2 (Isotonic linear functional). Let F (T ) be an algebra of real functions defined on T and L a subclass of F (T ) satisfying the axioms: (i) f, g ∈ L ⇒ f + g ∈ L; (ii) f ∈ L, α ∈ R ⇒ αf ∈ L. A functional A defined on L is an isotonic linear functional on L provided that: Proof : To prove the result in corollary 2(a), see for example, author in [11]. We prove the finding in corollary 2(b) by setting f (t)g(t) ≥ 1, then .
Applying the second quotient inequality to the right hand side of the above inequality yields This completes the proof.

Using the quotient Inequalities to obtain the estimate of Hahn-Banach contraction mapping theorem
In this section, the first product inequality is used to obtain an alternative way for proving Hahn Banach contraction mapping theorem. Proposition 1. A contraction mapping T , defined on a complete normed linear space, has unique fixed point.
Proof : Setting a mapping T : X → X, and let x o ∈ X such that T x n−1 = x n , ∀ n = 1, 2, . . . .
For any positive integer n, then where, α, is the boundedness constant. Using the first quotient inequality on the right hand side of the above inequality yields We can see that (1−α) α n → 0 as n → ∞ for all α ≥ 1. The sequence {X n } ∞ n=1 is convergent. The normed space X is complete since {X n } ∞ n=1 has a limit point in X. Let x be the element of X such that lim n→∞ X n = x. Thus, By the continuity of T . We can see that: Suppose further that T y 1 = y 1 and T y 2 = y 2 . Then which is a contradiction. Thus, the fixed point theorem is unique.

Using the Product and quotient Inequalities to obtain sharp Inequalities in Sobolev spaces
Lemma 1. For any 1 ≤ p < n, W 1,p (R n ) → L r (R n ) is continuously imbedded, ∀ α ∈ (0, 1] and u ≥ 2. Proof :Setting α ∈ (0, 1] and u ∈ W 1,p (R n ), we see that u ∈ L p * (R n ). Then Using the second product inequality, we obtain This completes the proof.

The Applications of the First Quotient Inequality to Unitary Space
In this section, the estimates involving the quotients of norms in the unitary space are obtained by using both the first and second product inequalities.
Corollary 3. Suppose that (X, · q ) is a continuously quasi-normed space. Then there exists 0 < p < ∞, such that whenever x and y are in X with δ = δ(x, y) > 0, then Proof : We see that: 2π 0 x p re iθ y p dθ

Quotient Inequalities involving Powers
In this section, the inequalities involving the relationships among index product of numbers and its quotient of numbers as bases are established for given regions of validity. Thus, we introduce in this paper the index power quotient inequalities; the first and second index power quotient inequalities. The main result is expressed in theorem (7), which is the fundamental tool for proving other results in this paper. involving propositions and corollaries.
Theorem 7 (First index power quotient inequality). Let T : X → Y be a Banach space. Then x pq ≤ x p q , ∀ x ∈ X, 0 ≤ p < ∞, and 0 < q ≤ 1, where equality occurs at either p = 0 or q = 1. Proof : Setting 0 ≤ p < ∞, and 0 < q ≤ 1. We can see that: This completes the proof. The converse of theorem 7 is stated in theorem 8 below.
Proof : The proof of theorem 8 is similar to theorem 7.
Theorem 9. Let T : X → Y be a Banach space. Then a b ≤ pq 2 a p + p 2 q b q , ∀ a, b ∈ X p, q ≤ 1.
Proof : Using the Young's inequality, we have ab ≤ qa p + pb q pq a b ≤ qa p + pb q pq .