Inequalities for a generalized finite Hilbert transform of convex functions

In this paper we obtain some new inequalities for a generalized finite Hilbert transform of convex functions. Applications for particular instances of finite Hilbert transforms are given as well.


Introduction
Finite Hilbert transform on the open interval (a, b) is defined by for t ∈ (a, b) and for various classes of functions f for which the above Cauchy Principal Value integral exists, see [14,Section 3.2] or [18, Lemma II. 1.1]. Suppose that I is an interval of real numbers with interiorI and f : I → R is a convex function on I. Then f is continuous onI and has finite left and right derivatives at each point ofI. Moreover, if x, y ∈I and x < y, then f − (x) ≤ f + (x) ≤ f − (y) ≤ f + (y) which shows that both f − and f + are nondecreasing function onI. It is also known that a convex function must be differentiable except for at most countably many points.
For a convex function f : I → R, the subdifferential of f denoted by ∂f is the set of all functions ϕ : I → [−∞, ∞] such that ϕ I ⊂ R and f (x) ≥ f (a) + (x − a) ϕ (a) for any x, a ∈ I.
It is also well known that if f is convex on I, then ∂f is nonempty, f − , f + ∈ ∂f and if ϕ ∈ ∂f , then f − (x) ≤ ϕ (x) ≤ f + (x) for any x ∈I.
In particular, ϕ is a nondecreasing function. If f is differentiable and convex onI, then ∂f = {f } .
The following result holds for the finite Hilbert transform of convex functions.
We can naturally generalize the concept of Hilbert transform as follows. For a continuous strictly increasing function g : [a, b] → [g (a) , g (b)] that is differentiable on (a, b) we define the following generalization of the finite Hilbert transform of a function f : (a, b) → C by for t ∈ (a, b) , provided the above P V exists. For [a, b] ⊂ (0, ∞) and g (t) = ln t, t ∈ [a, b] we have the logarithmic finite Hilbert transform defined by (2) ( where t ∈ (a, b) .

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Inequalities for a generalized finite Hilbert transform Similar transforms can be associated to the following functions as well: where ρ > 0, and Motivated by the above results, we establish in this paper some inequalities for the generalized finite Hilbert transform of convex functions on an interval. Applications for some particular instances of finite Hilbert transforms such as the ones from (2)-(7) are given as well.

Main Results
Consider the function 1 (t) = 1, t ∈ (a, b). We need the following preliminary result.

Lemma 1. For a continuous strictly increasing function
We also have for f : for t ∈ (a, b) , provided that the PV from the right hand side of the equality (9) exists.
Proof. We have for t ∈ (a, b) .
Since g is differentiable, we have for t ∈ (a, b), and by (10) we get (8).
From the definition (1) we deduce for t ∈ (a, b), which proves the identity (9).
If g is a function which maps an interval I of the real line to the real numbers, and is both continuous and injective then we can define the gmean of two numbers a, b ∈ I as 2 .
If I = R and g (t) = t is the identity function, then the LogMeanExp function.
] is a continuous strictly increasing function that is differentiable on (a, b) , f a function such that f • g −1 : (g (a) , g (b)) → R is a convex function on (g (a) , g (b)). Then for t ∈ (a, b) we have .
In particular, we have a, b)) .
Proof. For t, τ ∈ (a, b) with t = τ we then have .
By the convexity of f • g −1 we can state that for all g (a) ≤ c < d ≤ g (b) we have Since f • g −1 has lateral derivatives for z ∈ (g (a) , g (b)) it follows f has lateral derivatives in each point of (a, b) and by the chain rule and the derivative of the inverse function, (14) f Let t ∈ (a, b) and t − a > ε > 0, then by (13)) and (14) we have for τ ∈ (a, t − ε) . If we integrate the inequality (15) over τ on (a, t − ε), we get by (12) that for t ∈ (a, b) and t − a > ε > 0.
If we integrate the inequality (22) over τ on (t + ε, b), we get for t ∈ (a, b) and b − t > ε > 0. By adding the inequalities (21) and (23) we get for t ∈ (a, b) and min {b − t, t − a} > ε > 0. By taking the limit over ε → 0+ in (24) we get for t ∈ (a, b) . By using the identity (9) we obtain the second inequality in (11).

Remark 1.
With the assumptions of Theorem 2, and if f is differentiable on (a, b) , then we have for all t ∈ (a, b) .
In particular, we have a, b)) .
We also have: ] is a continuous strictly increasing function that is differentiable on (a, b) and g + (a) and g − (b) are finite. If f • g −1 : (g (a) , g (b)) → R is a convex function on (a, b) and f has finite lateral derivatives f + (a) and f − (b) , then for t ∈ (a, b) we have In particular, for t = M g (a, b) we get Proof. We recall that if Φ : I → R is a continuous convex function on the interval of real numbers I and α ∈ I then the divided difference function Using this property for the function Φ : for any τ ∈ (c, d) , τ = t. By the gradient inequality for the convex function Φ we also have and for t ∈ (c, d) .