R E S E A R C H
Open Access
Certain inequalities involving the
k
-Struve
function
Kottakkaran Sooppy Nisar
1, Saiful Rahman Mondal
2and Junesang Choi
3**Correspondence:
3Department of Mathematics,
Dongguk University, Gyeongju, 38066, Republic of Korea Full list of author information is available at the end of the article
Abstract
We aim to introduce ak-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 thisk-Struve function.
MSC: 33C10; 26D07
Keywords: k-Struve function;k-gamma function;k-beta function;k-digamma function; Turán-type inequalities
1 Introduction and preliminaries
Díaz and Pariguan [] introduced and investigated the so-calledk-gamma function
k(x) := ∞
tx–e–t
k
k dt (x) > ;k∈R+. ()
Here and in the following, letC,R,R+,N, andZ– be the sets of complex numbers, real
numbers, positive real numbers, positive integers, and negative integers, respectively, and letN:=N∪ {}. For various properties of thek-gamma function and its applications to
generalize other related functions such ask-beta function andk-digamma function, we refer the interested reader, for example, to [–] and the references cited therein.
Nantomah and Prempeh [] defined thek-digamma functionk:=k/kwhose series
representation is given as follows:
k(t) :=logk–γ k –
t+
∞
n=
t nk(nk+t)
k∈R+;t∈C\kZ–, ()
whereγ is the Euler-Mascheroni constant (see, e.g., [], Section .). A calculation yields
k(t) = ∞
n=
(nk+t)
k,t∈R+. ()
Clearly,k(t) is increasing on (,∞).
Turán [] proved that the Legendre polynomialsPn(x) satisfy the following determinant
inequality:
Pn(x) Pn+(x)
Pn+(x) Pn+(x)
≤ (–≤x≤;n∈N), ()
where the equality occurs only whenx=±. Recently, many researchers have applied the above classical inequality () in various polynomials and functions such as ultraspheri-cal 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 followingk-Struve function (cf. [], p. , Entry ..):
Skν,c(x) := ∞
r=
(–c)r
k(rk+ν+k )(r+
)
x
r+νk+
. ()
Then we investigate thek-Struve function () as follows: We establish certain inequalities involvingSkν,c, one of which is shown to be related to the Turán-type inequality; we show that thek-Struve function satisfies a second-order non-homogeneous differential equa-tion; and we present an integral representation and recurrence relations for thek-Struve function.
2 Inequalities
The modifiedk-Struve function is given as
Lk
ν(x) :=S
k
ν,–(x), ()
which is normalized and denoted byLk
νas follows:
Lk
ν(x) =
x
ν k
k
ν+k
Lk
ν(x) = ∞
r=
fr(ν,k)xr+, ()
where
fr(ν,k) :=
k(ν+k ) k(rk+ν+k
)(r+ )r+
.
Here, we investigate monotonicity and log-convexity involvingLk
ν. To do this, we recall some known useful properties which are given in the following lemma (see []).
Lemma Consider the power series f(x) = ∞k=akxkand g(x) = ∞k=bkxk,where ak∈R and bk∈R+(k∈N).Further suppose that both series converge on|x|<r.If the sequence
{ak/bk}k≥is increasing(or decreasing),then the function x →f(x)/g(x)is also increasing
(or decreasing)on(,r).
If both f and g are even,or both are odd functions,then the above results will be appli-cable.
(i) Forν≥μ> –k/,then the functionx →Lk
μ(x)/L
k
ν(x)is increasing onR.
(ii) The functionν →Lk
ν(x)is decreasing for fixedx∈[,∞)and increasing for fixed
x∈(–∞, ).Also,the functionν →Lk
ν(x)is log-convex on(–k/,∞)for fixed
x∈R+.
(iii) The functionν →Lk
ν+k(x)/L k
ν(x)is decreasing on(–k/,∞)for fixedx∈R+.
Proof To prove (i), recall the series in (). Clearly,
Lk
ν(x) Lk
μ(x) =
∞
r=fr(ν,k)xr+
∞
r=fr(μ,k)xr+
.
Denotewr:=fr(ν,k)/fr(μ,k). Then
wr=
k(ν+
k)k(rk+μ+ k) k(rk+ν+k)k(μ+k).
Appealing to relation (), we find
wr+
wr
=k(ν+
k)k(rk+k+μ+ k) k(rk+k+ν+k)k(μ+k)
k(μ+
k)k(rk+ν+ k) k(rk+μ+k)k(ν+k)
=(rk+μ+
k)(rk+μ+
k)k(rk+ν+ k)
(rk+ν+k)(rk+ν+k)k(rk+μ+k) =
rk+μ+k
rk+ν+k ≤,
whose last inequality is valid from the conditionν≥μ> –k/. Finally, the result (i) fol-lows from Lemma .
For (ii), since ν> –k, we first observe the coefficientsfr(ν,k) > for allr∈N. Then
the logarithmic derivative offr(ν,k) with respect toνis
fr(ν,k)
fr(ν,k)
=k
ν+ k
–k
rk+ν+ k
≤,
whose last inequality follows from (). Sincefr(ν,k) > (r∈N; ν> –k),fr(ν,k)≤
(r∈N; ν> –k). Henceν →fr(ν,k) is decreasing on (–k/,∞). This implies that, for
μ≥ν> –k/,
∞
r=
fr(ν,k)xr+≥
∞
r=
fr(μ,k)xr+
x∈[,∞)
and
∞
r=
fr(ν,k)xr+≤
∞
r=
fr(μ,k)xr+
x∈(–∞, ).
This proves the first statement of (ii). In view of (), we have
∂ ∂ν
logfr(ν,k)
=
∞
n=
(nk+ν+
k)
–
(nk+rk+ν+ k)
for allk∈R+and ν> –k. Thereforeν →fr(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
d dx
Lk
μ(x) Lk
ν(x)
≥, ()
for allx∈R+andν≥μ> –k/. In view of relation (), () is equivalent to
x–μkLk μ(x)
x–kνLk ν(x)
–x–μkLk μ(x)
x–νkLk ν(x)
≥ ()
for allx∈R+andν≥μ> –k/.
Considering () and settingc= – in () gives
d dx
x–kνLk ν(x)
=
–νk
√
π k(ν+3k )
+x–νkLk
ν+k(x). ()
Applying () to inequality () and using (), we obtain
x–μ+kνLk μ+k(x)L
k
ν(x) –L
k
ν+k(x)L k
μ(x)
≥ –
ν k
√
π k(ν+3k )
x–μkLk μ(x) –
–μk
√
π k(μ+3k )
x–νkLk ν(x)
=
–μ+kν
√
π k(μ+3k )k(ν+
3k )
Lk
μ(x) –L
k
ν(x)
≥ ()
for allx∈R+andν≥μ> –k/. Here, the last inequality in () follows from the first
statement of (ii). Also, we find from () that
Lk
μ+k(x) Lk
μ(x) –L
k
ν+k(x) Lk
ν(x)
≥
for allx∈R+andν≥μ> –k/. This proves (iii).
Remark One of the most significant consequences of Theorem is the Turán-type
in-equality for the functionLk
ν. The log-convexity ofLkν (the last statement of (ii) in Theo-rem ) implies
Lk
αν+(–α)ν(x)≤
Lk
ν
α (x)Lk
ν
–α (x)
α∈[, ];x,k∈R+;ν,ν∈(–k/,∞)
. ()
Choosingα= / and settingν=ν–aandν=ν+afor somea∈Rin () yields the
following reversed Turán-type inequality (cf. ()):
Lk
ν(x)
–Lk
ν–a(x)L k
ν+a(x)≤
3 Formulae for thek-Struve function
Here, we present a differential equation and recurrence relations regarding thek-Struve functionSkν,c().
Proposition Letk∈R+andν> –k.Then thek-Struve functionSk
ν,c()satisfies the
following second-order non-homogeneous differential equation:
xd
y
dx+x
dy dx+
k
ckx–νy= (
x
)
ν k+
kk(ν+k )(
)
. ()
Proof By using thek-Struve functionSk
ν,c() and the functional relation
k(x+k) =xk(x), ()
we find
x d
dxS k
ν,c(x) +x
d dxS
k
ν,c(x)
= ∞
r=
(–c)r(r+ν
k+ )
k(rk+ν+k )(r+
)
x
r+νk+
= ∞
r=
(–c)r(r+ )(r+ν
k + ) k(rk+ν+k
)(r+ )
x
r+kν+
+ν
kS k
ν,c(x)
= k
∞
r=
(–c)r(r+
)(rk+ν+ k ) k(rk+ν+k
)(r+ )
x
r+kν+
+ν
kS k
ν,c(x)
= k
(x)νk+
k(ν+k )( ) + k ∞ r=
(–c)r
k(rk+ν+k )(r+
)
x
r+kν+
+ν
kS k
ν,c(x)
= k
(x)νk+
k(ν+k )( ) –cx k S k
ν,c(x) +
ν
kS k
ν,c(x).
This shows thaty=Skν,c(x) the differential equation ().
Theorem Letk∈R+andν> –k.Then the following recurrence relations hold true:
d dx
xνkSk ν,c(x)
=
kx ν kSk
ν–k(x); ()
d dx
x–kνSk ν,c(x)
=
–νk
√
π k(ν+3k )
–cx–νkSk
ν+k,c(x); ()
kS
k
ν–k(x) –cS k
ν+k,c(x) =
d dxS
k
ν,c(x) –
(x/)kν
√
π k(ν+3k )
; ()
kS
k
ν–k(x) +cS k
ν+k,c(x) =
ν xkS
k
ν,c(x) +
(x/)νk
√
Proof From () we have
xνkSk ν,c(x) =
∞
r=
(–c)r
k(rk+ν+k )(r+
)
xr+kν+ r+kν+ ,
which, upon differentiating with respect toxand using relation (), yields
d dx
xνkSk ν,c(x)
=
∞
r=
(–c)r(r+ν
k + ) k(rk+ν+k
)(r+ )
xr+kν r+kν+ =x
ν k
k ∞
r=
(–c)r
k(rk+ν+k)(r+)
x
r+νk
= kx
ν kSk
ν–k,c(x).
This proves (). We can establish the result () by a similar argument as in the proof of (). We omit the details.
Similarly, from (), we obtain
x d dxS
k
ν,c(x) +
ν
kS
k
ν,c(x) =
x
kS
k
ν–k(x) ()
and
x d dxS
k
ν,c(x) –
ν
kS
k
ν,c(x) =
x(x/)νk
√
π k(ν+3k )
–cxSkν+k,c(x). ()
Adding and subtracting each side of () and () yields, respectively, the results ()
and ().
4 Integral representations
Here, we present two integral representations for the functionSk
ν,c.
Theorem Letk∈R+,(ν) > –k
,andα∈R\ {}.Then
Skν,α(x) =
√k
α√π k(ν+k )
x
ν
k
–tνk–sin
αxt
√
k
dt ()
and
Skν,–α(x) =
√k
√
π k(ν+k )
x
ν
k
–tkν–
sinh
αxt
√
k
dt. ()
In particular,we have
–cos
αx
√
k
=α k
πx
S
k
,α(x) ()
and
cosh
αx
√
k
– =α k
πx
S
k
ν
Proof We begin by recalling thek-beta function (see [])
Bk(x,y) =
k(x)k(y)
k(x+y) = k
txk–( –t)
y
k–dt
k∈R+;min(x),(y)> . ()
Replacingtbyton the right-hand-sided integral in (), we obtain
Bk(x,y) =
k
tkx– –t
y
k–dt. ()
Settingx= (r+ )kandy=ν+k/ in () gives
k(rk+ν+k)=
k((r+ )k)k(ν+k )
tr –tkν–
dt. ()
Applying the known identity
k(kx) =kx–(x) ()
and the Legendre duplication formula (see [, , ])
(z)z+
= –z√π (z) ()
to the functionSkν,cwith (), we get
Skν,c(x) =
√
k
√
π k(ν+k )
x
ν
k
–t
ν k–
∞
r=
(–c)r
(r+ )!
xt
√
k r+
dt. ()
Finally, settingc=±α(α∈R\{}) in () yields, respectively, the desired results () and
(). Further, settingν=k/ in () and () yields, respectively, the desired results ()
and ().
5 Results and discussion
We introduce ak-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, whose many variants have been investigated. We also present a differential equation, several recurrence relations, and integral representations for thisk-Struve function.
6 Conclusions
The results presented here are sure to be new and potentially useful. Since the research subject here and its related ones are competitive, the content of this paper may attract interested readers who have been interested in this and related research subjects.
Competing interests
Authors’ contributions
The authors have contributed equally to this manuscript. They read and approved the final manuscript.
Author details
1Department of Mathematics, College of Arts and Science-Wadi Aldawaser, Prince Sattam bin Abdulaziz University, Wadi
Ad-Dawasir, Riyadh region 11991, Saudi Arabia.2Department of Mathematics and Statistics, College of Science, King
Faisal University, Hofuf, Al-Hasa 31982, Saudi Arabia.3Department of Mathematics, Dongguk University, Gyeongju, 38066,
Republic of Korea.
Acknowledgements
The authors would like to express their deep-felt gratitude for the reviewer’s detailed reviewing and useful comments.
Publisher’s Note
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Received: 27 November 2016 Accepted: 24 March 2017
References
1. Díaz, R, Pariguan, E: On hypergeometric functions andk-Pochhammer symbol. Divulg. Mat.15(2), 179-192 (2007) 2. Nantomah, K, Prempeh, E: Some inequalities for thek-digamma function. Math. Æterna4(5), 521-525 (2014) 3. Mubeen, S, Naz, M, Rahman, G: A note onk-hypergemetric differential equations. J. Inequal. Spec. Funct.4(3), 8-43
(2013)
4. Srivastava, HM, Choi, J: Zeta andq-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam (2012) 5. Turán, P: On the zeros of the polynomials of Legendre. ˇCas. Pˇest. Math. Fys.75, 113-122 (1950)
6. Karlin, S, Szegö, G: On certain determinants whose elements are orthogonal polynomials. J. Anal. Math.8, 1-157 (1960)
7. Abramowitz, M, Stegun, IA (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Tenth Printing. National Bureau of Standards, Applied Mathematics Series, vol. 55. Natl. Bur. of Standards, Washington (1972). Reprinted by Dover Publications, New York, 1965 (see also [10])
8. Biernacki, M, Krzy˙z, J: On the monotonicity of certain functionals in the theory of analytic functions. Ann. Univ. Mariae Curie-Skłodowska, Sect. A9, 135-147 (1957)
9. Andrews, GE, Askey, R, Roy, R: Special Functions. Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge University Press, Cambridge (1999)
