Certain inequalities involving the k-Struve function

We aim to introduce a k-Struve function and investigate its various properties, including mainly certain inequalities associated with this function. One of the inequalities given here is pointed out to be related to the so-called classical Turán-type inequality. We also present a differential equation, several recurrence relations, and integral representations for this k-Struve function.


Introduction and preliminaries
introduced and investigated the so-called k-gamma function Here and in the following, let C, R, R + , N, and Zbe the sets of complex numbers, real numbers, positive real numbers, positive integers, and negative integers, respectively, and let N  := N ∪ {}. For various properties of the k-gamma function and its applications to generalize other related functions such as k-beta function and k-digamma function, we refer the interested reader, for example, to [-] and the references cited therein. Nantomah and Prempeh [] defined the k-digamma function k := k / k whose series representation is given as follows: where γ is the Euler-Mascheroni constant (see, e.g., [], Section .). A calculation yields Clearly, k (t) is increasing on (, ∞).
Turán [] proved that the Legendre polynomials P n (x) satisfy the following determinant inequality: where the equality occurs only when x = ±. Recently, many researchers have applied the above classical inequality () in various polynomials and functions such as ultraspherical polynomials, Laguerre polynomials, Hermite polynomials, Bessel functions of the first kind, modified Bessel functions, and polygamma functions. Karlin and Szegö [] named such determinants as in () Turánians.
In this paper, we consider the following k-Struve function (cf. [], p. , Entry ..): Then we investigate the k-Struve function () as follows: We establish certain inequalities involving S k ν,c , one of which is shown to be related to the Turán-type inequality; we show that the k-Struve function satisfies a second-order non-homogeneous differential equation; and we present an integral representation and recurrence relations for the k-Struve function.

Inequalities
The modified k-Struve function is given as which is normalized and denoted by L k ν as follows: Here, we investigate monotonicity and log-convexity involving L k ν . To do this, we recall some known useful properties which are given in the following lemma (see []).

Lemma  Consider the power series f
If both f and g are even, or both are odd functions, then the above results will be applicable.
Theorem  Let k ∈ R + be fixed. Then the following statements hold.
Proof To prove (i), recall the series in (). Clearly, .
Appealing to relation (), we find whose last inequality is valid from the condition ν ≥ μ > -k/. Finally, the result (i) follows from Lemma . For (ii), since ν > -k, we first observe the coefficients f r (ν, k) >  for all r ∈ N  . Then the logarithmic derivative of f r (ν, k) with respect to ν is whose last inequality follows from (). Since f r (ν, k) >  (r ∈ N  ; ν > -k), f r (ν, k) ≤  (r ∈ N  ; ν > -k). Hence ν → f r (ν, k) is decreasing on (-k/, ∞). This implies that, for μ ≥ ν > -k/, This proves the first statement of (ii). In view of (), we have for all k ∈ R + and ν > -k. Therefore ν → f r (ν) is log-convex on (-k/, ∞). Since a sum of log-convex functions is log-convex, the second statement of (ii) is proved. For (iii), it is obvious from (i) that for all x ∈ R + and ν ≥ μ > -k/. In view of relation (), () is equivalent to for all x ∈ R + and ν ≥ μ > -k/. Considering () and setting c = - in () gives Applying () to inequality () and using (), we obtain for all x ∈ R + and ν ≥ μ > -k/. Here, the last inequality in () follows from the first statement of (ii). Also, we find from () that for all x ∈ R + and ν ≥ μ > -k/. This proves (iii).
Remark  One of the most significant consequences of Theorem  is the Turán-type inequality for the function L k ν . The log-convexity of L k ν (the last statement of (ii) in Theorem ) implies Choosing α = / and setting ν  = νa and ν  = ν + a for some a ∈ R in () yields the following reversed Turán-type inequality (cf. ()):

Formulae for the k-Struve function
Here, we present a differential equation and recurrence relations regarding the k-Struve function S k ν,c ().
Proposition  Let k ∈ R + and ν > -k. Then the k-Struve function S k ν,c () satisfies the following second-order non-homogeneous differential equation: .

(   )
Proof By using the k-Struve function S k ν,c () and the functional relation we find This shows that y = S k ν,c (x) the differential equation ().