2. Inequalities of the Tur´ an type

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DOI: 10.24193/subbmath.2019.1.06

Some inequalities of the Tur´ an type for confluent hypergeometric functions of the second kind

Feng Qi, Ravi Bhukya and Venkatalakshmi Akavaram

Abstract. In the paper, by virtue of the Hölder integral inequality, the authors derive some inequalities of the Turán type for confluent hypergeometric functions of the second kind, for the Mellin transforms, and for the Laplace transforms, and improve some known inequalities of the Turán type.

Mathematics Subject Classification (2010):26D15, 26D20, 26D99, 33C15, 44A10, 44A15.

Keywords:Inequality of the Tur´an type, confluent hypergeometric function of the second kind, improvement, Mellin transform, Laplace transform, H¨older integral inequality.

1. Introduction

In 1950, P. Tur´an [16] proved that the Legendre polynomialsP_n(x) satisfy Pn(x)Pn+2(x)−P_n+1² (x)≤0

for|x| ≤1 andn= 0,1,2, . . ., where the equality holds only ifx=±1. An inequality of this kind is known as an inequality of the Tur´an type. This classical inequality has been extended to various special functions. For recent development on this classical inequality, please refer to [2, 3, 4, 6, 7, 11] and closely related reference therein.

It is known [1, p. 504-505] that confluent hypergeometric functions of the second kindψ(a, c, x) are also known as the Tricomi confluent hypergeometric functions, are a special solution of Kummer’s differential equation

xy⁰⁰(x) + (c−x)y⁰(x)−ay(x) = 0, and have the integral representation

ψ(a, c, x) = 1 Γ(a)

Z ∞

0

t^a−1(1 +t)^c−a−1e^−xtdt (1.1)

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fora >0,c∈R, andx >0, where Γ(z) =

Z ∞

0

t^z−1e^−tdt, <(z)>0 is the classical Euler gamma function [12, 13, 14, 15].

The Laplace transform and the Mellin transform of a function f(t) are respec- tively defined by

L(s) =L(f)(s) = Z ∞

0

f(t)e^−stdt, s >0 and

M(s) =M(f)(s) = Z ∞

0

f(t)t^s−1dt.

These transforms are widely-used integral transforms with many applications in physics and engineering.

In this paper, we will study some Tur´an type inequalities for confluent hypergeometric functions of the second kindψ(a, c, x).

2. Inequalities of the Tur´ an type

We are now in a position to find some inequality of the Tur´an type for confluent hypergeometric functions of the second kind. These newly-founded inequalities improve existed inequality of the Tur´an type in [4, Theorem 2].

Theorem 2.1. Forx >0,a >0, andc∈R, we have ψ²(a+ 1, c, x)< a+ 1

a ψ(a, c, x)ψ(a+ 2, c, x). (2.1)

Proof. The equality (1.1) can be reformulated as f(a),ψ(a, c, x)Γ(a) =

Z ∞

0

t 1 +t

^a

(1 +t)^c−1e^−xt1 t dt.

Replacing a by αp+ (1−α)q for p, q > 0, p 6= q, and α ∈ (0,1) and using the well-known H¨older integral inequality give

f(αp+ (1−α)q) = Z ∞

0

t 1 +t

αp+(1−α)q

(1 +t)^c−1e^−xt1 tdt

= Z ∞

0

t^p−1(1 +t)^c−p−1e^−xtα

t^q−1(1 +t)^c−q−1e^−xt1−α

<

Z ∞

0

t^p−1(1 +t)^c−p−1e^−xt

^αZ ∞

0

t^q−1(1 +t)^c−q−1e^−xt ^1−α

=f^α(p)f^1−α(q).

This implies that the function f is strictly logarithmically convex on (0,∞). Conse- quently, takingα= ¹₂,p=a, and q=a+ 2 leads tof²(a+ 1)< f(a)f(a+ 2) which is equivalent to (2.1). The proof of Theorem 2.1 is complete.

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Theorem 2.2. Forx >0,0< a1, a2< a, andc∈R, we have ψ²(a, c, x)<Γ(a1)Γ(a2)

Γ²(a) ψ(a1, c, x)ψ(a2, c, x). (2.2) Proof. We continue to adopt the notationf(a) in the proof of Theorem 2.1. Then

f⁰(a) = d da

Z ∞

0

t 1 +t

^a

(1 +t)^c−11 te^−xtdt

= Z ∞

0

t 1 +t

^a ln

t 1 +t

(1 +t)^c−11 te^−xtdt

<0.

Since ln _1+t^t

<0 for t >0, the functionf(a) is decreasing on (0,∞) with respect to a. Accordingly, for 0 < a1, a2 < a, we have f(a) < f(a1) and f(a) < f(a2).

Consequently, it follows that f²(a) < f(a1)f(a2) which is equivalent to (2.2). The

proof of Theorem 2.2 is complete.

Theorem 2.3. Forx >0,a >0, andc1, c2< c∈R, we have

ψ(a, c, x)ψ(a, c−1, x)< ψ²(a, c+ 1, x)< ψ(a, c, x)ψ(a, c+ 2, x) (2.3) and

ψ(a, c₁, x)ψ(a, c₂, x)< ψ²(a, c, x). (2.4) Proof. A straightforward computation yields

ψ⁰(c) = 1 Γ(a)

d dc

Z ∞

0

t^a−1(1 +t)^c−a−1e^−xtdt

= 1

Γ(a) Z ∞

0

t^a−1(1 +t)^c−a−1ln(1 +t)e^−xtdt

>0.

This means that the function ψ(a, c, x) is increasing with respect to c ∈ R. Hence, for c1, c2 < c, it follows that ψ(a, c1, x) < ψ(a, c, x) and ψ(a, c2, x) < ψ(a, c, x).

Consequently, we obtain the inequality (2.4).

Takingc1=c−1 andc2=c and replacingcbyc+ 1 in (2.4) deduce that ψ(a, c−1, x)ψ(a, c, x)< ψ²(a, c+ 1, x).

From this inequality and the inequality ψ²(a, c+ 1, x) < ψ(a, c, x)ψ(a, c+ 2, x) in the paper [4], the inequality (2.3) follows immediately. The proof of Theorem 2.3 is

complete.

Theorem 2.4. Forx, y >0,a >0,p, q >0 such that ¹_p+¹_q = 1, andc∈R, we have ψ

a, c,x

p+y q

< ψ^1/p(a, c, x)ψ^1/q(a, c, y). (2.5)

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Proof. Applying the well-known H¨older integral inequality to the third variablexin ψ(a, c, x) arrives at

ψ

a, c,x p+y

q

= 1

Γ(a) Z ∞

0

t^a−1(1 +t)^c−a−1e−(x/p+y/q)tdt

= 1

Γ(a) Z ∞

0

t^a−1(1 +t)^c−a−1e^−xt^1/p

t^a−1(1 +t)^c−a−1e^−yt^1/q

<

1 Γ(a)

Z ∞

0

t^a−1(1 +t)^c−a−1e^−xt

^1/p 1 Γ(a)

Z ∞

0

t^q−1(1 +t)^c−q−1e^−yt ^1/q

=ψ^1/p(a, c, x)ψ^1/q(a, c, x).

Therefore, the inequality (2.5) is proved. The proof of Theorem 2.4 is complete.

Theorem 2.5. Forx, y >1such that ¹_x+¹_y ≤1,a >0,p, q >0 such that ¹_p+¹_q = 1, andc∈R, we have

ψ

a, c,x^p p +y^q

q

< ψ^1/p(a, c, px)ψ^1/q(a, c, qy). (2.6) Proof. Applying Young’s inequality to the third variablexin ψ(a, c, x) results in

ψ x^p

p +y^q q

= 1

Γ(a) Z ∞

0

t^a−1(1 +t)^c−a−1e^−(x^p^/p+y^q^/q)tdt

≤ 1 Γ(a)

Z ∞

0

t^a−1(1 +t)^c−a−1e^−xyt

≤ 1

Γ(a) Z ∞

0

t^a−1(1 +t)^c−a−1e^−pxt 1/p

× 1

Γ(a) Z ∞

0

t^q−1(1 +t)^c−q−1e^−qyt 1/q

=ψ^1/p(a, c, px)ψ^1/q(a, c, qx).

The inequality (2.6) is thus proved. The proof of Theorem 2.5 is complete.

Theorem 2.6. Forx, y >0,a >0, andc∈R, we have

ψ²(a, c, x+y)< ψ(a, c, x)ψ(a, c, y). (2.7) Forx >0,0< y <1,a >0, andc∈R, we have

ψ(a, c, x+y)< ψ(a, c, xy). (2.8)

Proof. It is easy to see that the functionψ(a, c, x) is decreasing with respect tox∈ (0,∞). Since x < x+y and y < x+y, it follows thatψ(a, c, x+y)< ψ(a, c, x) and ψ(a, c, x+y)< ψ(a, c, y). This means the inequality (2.7).

Similarly, the inequality (2.8) follows readily. The proof of Theorem 2.6 is complete.

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3. Inequalities of the Tur´ an type for the Mellin transform

Now we discover an inequality of the Tur´an type for the Mellin transform.

Theorem 3.1. Fors >0, the Mellin transformM(s) satisfies

F²(s+ 1)≤F(s)F(s+ 2). (3.1)

Proof. Applying the H¨older integral inequality finds that M(αp+ (1−α)q) =

Z ∞

0

f(t)tαp+(1−α)q−1dt

= Z ∞

0

f(t)t^p−1α

f(t)t^q−11−α

dt

≤ Z ∞

0

f(t)t^p−1dt

αZ ∞

0

f(t)t^q−1dt 1−α

=M^α(p)M^1−α(q).

This means that the Mellin transform M(s) is strictly logarithmically convex on (0,∞). Further letting α= ¹₂, p=s, andq=s+ 2 in the above inequality leads to the inequality (3.1). The proof of Theorem 3.1 is complete.

Example 3.2. Entry 17.43.26 in [9] states that

M1(s) =M(cosech(x)) = 2(1−2^−s)Γ(s)ζ(s), s >1.

By Theorem 3.1, it follows readily that

M₁²(s+ 1)≤M1(s)M1(s+ 2).

After some simplification we acquire ζ²(s+ 1)≤

s+ 1 s

(1−2^−s)(1−2^−s−2) (1−2^−s−1)²

ζ(s)ζ(s+ 2), s >1 which improves the Tur´an type inequality for the zeta function in [11].

Example 3.3. Entry 6.3.8 in [10] states that M2(s) =M e^−ax(1−e^−x)⁻¹

= Γ(s)ζ(s, a), s >0, a >0.

By Theorem 3.1, we derive

M₂²(s+ 1)≤M₂(s)M₂(s+ 2).

After some simplification we acquire ζ²(s+ 1, a)≤s+ 1

s ζ(s, a)ζ(s+ 2, a), s >1, a >0. (3.2) Whena= 1 in (3.2), we recover the Tur´an type inequality in [11].

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4. Inequalities of the Tur´ an type for the Laplace transform

Finally we find out an inequality of the Tur´an type for the Laplace transform.

Theorem 4.1. The Laplace transformL(s)satisfies

L²(s+ 1)≤L(s)L(s+ 2), s >0. (4.1)

Proof. By the H¨older integral inequality, we have L(αp+ (1−α)q) =

Z ∞

0

f(t)e−(αp+(1−α)q)tdt

= Z ∞

0

f(t)e^−ptα

f(t)e^−qt1−α

dt

≤ Z ∞

0

f(t)e^−ptdt

^αZ ∞

0

f(t)e^−qtdt 1−α

=L^α(p)L^1−α(q).

In other words, the Laplace transformL(s) is strictly logarithmically convex on (0,∞).

Specially, setting α= ¹₂,p=s, and q=s+ 2 in the above inequality leads to (4.1).

The proof of Theorem 4.1 is complete.

Example 4.2. Entry 4.15.29 in [10] states that L3(s) =L (1−e^−t)^ν/2Jν(a(1−e^−t)^1/2)

= Γ(s) 2

a s

Jν+s(a)

fors >0,a >0,ν >−1, whereJµ(z) denotes Bessel’s functions. By Theorem 4.1, it follows that

L²₃(s+ 1)≤L₃(s)L₃(s+ 2) which can be reformulated as

J_ν+s+1² (a)≤ s+ 1

s J_ν+s(a)J_ν+s+2(a) (4.2)

fors >0, 1> a >0, andν >−¹₂.

When taking ν = 0 and replacing s by s−1 for s ≥ 1 in (4.2), we derive an upper bound of the Tur´an type inequality in [5, Eq. (2.3)] for 0< a <1.

Example 4.3. Entry 4.3.11 in [10] reads that L4(s) =L(t²−a²)^ν−1/2= 1

√πΓ

ν+1 2

2a s

^ν Kν(as)

fors, a >0 andν >−¹₂, where K_µ(z) denotes modified Bessel’s functions. By Theo- rem 4.1, it follows that

L²₄(s+ 1)≤L4(s)L4(s+ 2) which can be rewritten as

K_ν²(a(s+ 1))≤

s²+ 2s+ 1 s(s+ 2)

ν

Kν(as)Kν(a(s+ 2)) fors, a >0 andν >−¹₂.

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Remark 4.4. Many other Tur´an type inequalities can be obtained for functions whose Laplace and Mellin transforms exists. In particular, we can prove some Tur´an type inequalities for the gamma, beta, extended beta, hypergeometric, error, and compliment error functions.

Remark 4.5. This paper is a slightly revised version of the preprint [8].

Acknowledgements. The authors appreciate anonymous reviewers for their valuable comments on the original version of this paper.

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Feng Qi

(1) College of Mathematics, Inner Mongolia University for Nationalities Tongliao 028043, Inner Mongolia, China;

(2) School of Mathematical Sciences, Tianjin Polytechnic University Tianjin 300387, China;

(3) Institute of Mathematics, Henan Polytechnic University Jiaozuo 454010, Henan, China

e-mail:[email protected], [email protected] Ravi Bhukya

Department of Mathematics, Government College for Men Kurnool, Andhra Pradesh, India

e-mail:[email protected] Venkatalakshmi Akavaram

Department of Mathematics, University College of Technology Osmania University, Hyderrabad, Telangana, India

e-mail:[email protected]