Some new discrete Hilbert’s inequalities involving Fenchel–Legendre transform

Some new Hilbert-type inequalities involving Fenchel–Legendre transform are introduced. These inequalities give more general forms of some previously proved inequalities.

In [8] the author gave inequalities that can be considered as an extension to inequality (1), containing a series of positive terms as follows.
Theorem 1 Let q ≥ 1, p ≥ 1, and let (a n ) and (b m ) be two positive sequences of real numbers defined for n = 1, 2, . . . , k and m = 1, 2, . . . , r, where k, r ∈ N, and define A n = n s=1 a s , (kn + 1) A p-1 n a n In [7], the author gave an improvement of the inequality given in Theorem 1 as follows.
In this paper, through Fenchel-Legendre transform and by utilizing Jensen's and Schwarz's inequalities, we give some improvements of the inequalities given in Theorems 1 and 2. In addition, some new Hilbert-type inequalities are obtained alongside some applications.

Preliminaries
In this section we introduce the Fenchel-Legendre transform, which will have an important role in later sections. For more details, we refer, for instance, to [9][10][11].
Then the Fenchel-Legendre transform is defined as follows: where ·, · denotes the scalar product on R n . The mapping h − → h * will often be called the conjugate operation.
In addition, the domain of h * , i.e., dom(h * ) is the set of slopes of all the affine functions minorizing the function h over R n .
With more hypotheses on h we can give, in the next corollary, an equivalent formula for (5) called Legendre transform.

Main results
We begin this section by proving the following simple and useful lemma.

Lemma 2 For x and y ∈ R.
Assume that x + y ≥ 1, then Proof First, we use x + y ≥ 1 and α β ≥ 1 to write (x + y) and (a n ) 1≤n≤k , (b m ) 1≤m≤r be two positive sequences of real numbers where k, r ∈ N. Define A n = n s=1 a s , B m = m t=1 b t . Then the following inequalities hold: Proof By exploiting the following inequality [14,15] Using (11), (12), and the Schwarz inequality, we observe that squaring both sides of inequality (13) gives Using (6) (for nonnegative real numbers x and y) in (13) and (14) produces Let us divide both sides of (15) by (h(n) + h * (m)) 1 2 , take the sum over n from 1 to k afterwards and the sum over m from 1 to r subsequently. Besides, we use the Schwarz inequality, and then we interchange the order of the summations (see [14,15]). We obtain ≤ pq √ kr k n=1 a n A p-1 n 2 (kn + 1) Now apply Lemma 2 on L.H.S. of (17) to obtain (10). To prove (9), divide both sides of (16) by h(n) + h * (m), take the sum over n from 1 to k afterwards, then the sum over m from 1 to r, and then interchange the order of the summations to obtain unless (a n ) or (b m ) is null, where Proof By the hypothesis that √ n ∈ dom(h), √ m ∈ dom(h * ), inequality (6) gives Complete the proof as we did to obtain inequality (10) in Theorem 3 with appropriate changes. (a n ) 2 (kn + 1) hold.
The following theorem treats the further generalization of the inequality obtained in ≤ M 1 (k, r) k n=1 p n Φ a n p n 2 (kn + 1) where Proof Using the fact that Φ is a submultiplicative function, we have From inequalities (24), (25) and the Fenchel-Young inequality (for nonnegative reals x and y), we have Let us divide both sides of (26) by (h(n) + h * (m)) 1 2 , take the sum over n from 1 to k afterwards, then take the sum over m from 1 to r. Additionally, use the Schwarz inequality and then interchange the order of the summations to have Now define M 1 (k, r) as Now apply Lemma 2 on the L.H.S. of (28) to obtain (22). This completes the proof.

Lemma 3
Under the hypotheses of Theorem 5, the following inequality holds: p n Φ a n p n 4 (kn + 1) where M 2 (k, r) = k n=1 nΦ a n P n Proof From inequalities (24), (25) and the Fenchel-Young inequality (for nonnegative reals x and y), we have Now divide both sides of (30) by h(n) + h * (m), then take the sum over n from 1 to k first and the sum over m from 1 to r, then use the Schwarz inequality to obtain where M 2 (k, r) = k n=1 nΦ a n P n The following theorem deals with slight changes of the inequality given in Theorem 9.
Theorem 7 Let (a n ) 1≤n≤k , (b m ) 1≤m≤r , (p n ) 1≤n≤k , and (q m ) 1≤m≤r be nonnegative sequences of real numbers where k, r ∈ N. Suppose that Φ and Ψ are nonnegative, convex, and submultiplicative functions on [0, ∞). Let A n , B m be defined as follows: p n Φ(a n ) 2 (kn + 1) (37) similarly, The rest of the proof is similar to the proof of Theorems 3 and 5 with suitable changes.

Corollary 4
Under the hypotheses of Theorem 7, the following inequality holds: Proof To prove this result, take p s = q t = 1 for all s ≥ 1, t ≥ 1, then P n = n, Q m = m and use the fact that

Some applications
In this section we try to show the beauty behind our results. We achieve this by utilizing inequality (10)  (kn + 1) A p-1 n a n which is inequality (3) as desired.