On some inequalities involving Turán-type inequalities

It is well known that the Cauchy–Bunyakovsky–Schwarz inequality plays an important role in different branches of modern mathematics such as Hilbert space theory, classical real and complex analysis, numerical analysis, qualitative theory of differential equations and probability and statistics. To date, a large number of generalisations and refinements of this inequality have been investigated in the literature, e.g. (Alzer, 1999; Callebaut, 1965; Masjed-Jamei, 2009; Masjed-Jamei, Dragomir, & Srivastava, 2009; Steiger, 1969; Zheng, 1998). (1) ⎛


Introduction
The integral representation of well-known Cauchy-Bunyakovsky-Schwarz inequality (see, for instance, Mitrinović, Pečarić, & Fink, 1993) in the space of continuous real-valued functions C a, b , ℝ is given by: It is well known that the Cauchy-Bunyakovsky-Schwarz inequality plays an important role in different branches of modern mathematics such as Hilbert space theory, classical real and complex analysis, numerical analysis, qualitative theory of differential equations and probability and statistics. To date, a large number of generalisations and refinements of this inequality have been investigated in the literature, e.g. (Alzer, 1999;Callebaut, 1965;Masjed-Jamei, 2009;Masjed-Jamei, Dragomir, & Srivastava, 2009;Steiger, 1969;Zheng, 1998). (1)

PUBLIC INTEREST STATEMENT
In this paper, we prove inequalities involving Turántype inequalities for some special functions using a new form of the Cauchy-Bunyakovsky-Schwarz inequality. These inequalities play an important role in different branches of modern mathematics such as Hilbert space theory, classical real and complex analysis, numerical analysis, probability and statistics. Also, Turán-type inequalities have important applications in complex analysis, number theory, theory of mean values or statistics and control theory.
Also, the importance, in many fields of mathematics, of the inequalities of the type: n = 0, 1, 2, … is well known. They are named, by Karlin and Szegö, Turán-type inequalities because the first of this type of inequalities was proved by Turán, 1950). Laforgia and Natalini, 2006) used the following form of the Schwarz inequality (1): to establish some new Turán-type inequalities involving the special functions as gamma, polygamma functions and Riemann's zeta function. Here, f and g are non-negative functions of a real variable and m and n belong to a set S of real numbers, such that the involved integrals in Equation (3) exist.
In this context, we have the idea to replace u(t) and v(t) in (1) by g(t)h x (t)f (t) and g(t)h (2− )x (t)f (t), respectively, to introduce the following new inequality: in which α, ν, μ ∊ ℝ and g, h, f are real integrable functions, such that the involved integrals in Equation (4) exist.
The aim of this paper is to apply the inequality (4) for some well-known special functions in order to get inequalities involving Turán-type inequalities.

The results
In this section, we apply the inequality Equation (4) to prove inequalities involving Turán-type inequalities for n-th derivative of gamma function and the Remainder of the Binet's first formula for ln Γ(x), polygamma functions, exponential integral function, Abramowitz's function and modified Bessel function of second kind.

An inequality for the n-th derivative of gamma function
Theorem 2.1 For every real number x ∈ 0, ∞ , ∈ 0, 2 and for every integer ν, μ ≥ 1, such that + is even, it holds for the n-th derivative of gamma function: Proof The classical Euler gamma function is defined for x > 0 as: By differentiating Equation (5), we obtain, for n = 1, 2, 3, … (2) Equation (4), we get: By applying Equation (6) in the above inequality, the following result will eventually be obtained: ∀ ∈ 0, 2 , x > 0 and for every integer ν, μ ≥ 1, such that + is even.

An inequality for the polygamma function
Theorem 2.2 For every real number x ∈ 0, ∞ , ∈ 0, 2 and for every integer ν, μ ≥ 1, such that ν + μ is even, it holds for the polygamma functions: Proof As we know, the polygamma functions (n) (x) = d n (x) dx n , where n = 1, 2, 3, …, are defined as the n-th derivative of the Psi function ( , (x > 0) with the usual notation for the gamma function and has an integral representation (Nikiforov & Uvarov, 1988) as: Now, if g(t) = 1 1−e −t , h(t) = e −t and f (t) = t are substituted in inequality Equation (4) for a, b = [0, ∞), the following inequality is derived: By the definition Equation (9), this is equivalent to: x > 0 and for every integer ν, μ ≥ 1, such that ν + μ is even.

An inequality for the n-th derivative of the remainder of the Binet's first formula for ln (x)
Theorem 2.3 For every real number x ∈ 0, ∞ , ∈ 0, 2 and for every integer ν, μ ≥ 1, such that ν + μ is even, it holds for the n-th derivative of the remainder of the Binet's first formula for the logarithm of the gamma function, i.e. ln Γ(x): Proof Binet's first formula for ln Γ(x) is given by: For x > 0, where the function: is known as the remainder of the Binet's first formula for the logarithm of the gamma function; see (Abramowitz & Stegun, 1965). (11), we obtain, for every positive integer n ≥ 1.

An inequality for the Abramowitz's function
Theorem 2.5 For every real number x ≥ 0, ∈ 0, 2 and for every non-negative integer ν and μ, such that ν + μ is even, it holds for the Abramowitz function: Proof The Abramowitz's function (Abramowitz & Stegun, 1965) which has been used in many fields of physics, as the theory of the field of particle and radiation transform, is defined as: where n is a non-negative integer and x ≥ 0. (13) t n e −t 2 −xt −1 dt