Some inequalities for special functions

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R E S E A R C H

Open Access

Some inequalities for special functions

Ayman Shehata

*

*_{Correspondence:}

[email protected] Department of Mathematics, Faculty of Science, Assiut University, Assiut, 71516, Egypt

Department of Mathematics, College of Science and Arts in Unaizah, Qassim University, Qassim, 10363, Kingdom of Saudi Arabia

Abstract

This paper is motivated by an open problem of several inequalities for conﬂuent hypergeometric functions. We give some inequalities for simple Laguerre, Laguerre, pseudo-Laguerre, Shively’s pseudo-Laguerre, and Hermite functions of one and two variables.

MSC: 26D15; 30A10; 26D07

Keywords: conﬂuent hypergeometric; simple Laguerre; Laguerre; pseudo-Laguerre; Shively’s pseudo-Laguerre; Hermite functions

1 Introduction

Inequalities for the special functions appear infrequently in the literature. Some of these inequalities are closely related to those presented here. Inequalities for the ratio of con-ﬂuent hypergeometric functions are available in the literature. The formulas are very im-portant, as they include expansions for many transcendent expressions of mathematical physics in series of the classical orthogonal polynomials, Laguerre, Hermite functions,etc.

[–]. The developments bear heavily on the works of Luke [–]. The author has earlier studied the inequalities for Humbert functions []. This motivated us to obtain two-sided inequalities for simple Laguerre, Laguerre, pseudo-Laguerre, Shively’s pseudo-Laguerre, and Hermite functions of one and two variables through conﬂuent hypergeometric func-tions of one and two variables.

In [], the conﬂuent hypergeometric function is deﬁned as

F(a;c;x) =

k≥

(a)k

k!(c)k

xk, c> ,x> , (.)

where (a)kis the Pochhammer symbol or shifted factorial deﬁned as

(a)k=a(a+ )(a+ )· · ·(a+k– ) =

(a+k)

(a) ; k≥; (a)= .

We recall here the theorems in [, ], which will be used in the investigation that fol-lows.

Theorem .(Erdélyiet al.[]) If a,c> and <x< ,then we have the inequalities of the ratios of a conﬂuent hypergeometric function

 +a

cx<F(a;c;x) <  +

a

c x. (.)

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Theorem .(Bordelon [], Ross []) (i)Let c>a> and y>x> ,then

ex–y<F(a;c;x)

F(a;c;y)

< . (.)

(ii)Let d>c>a> and y>x> ,then

F(a;c;x)

F(a;d;y)

>(c)(d–a)

(d)(c–a). (.)

Theorem .(Joshi and Bissu []) (i)Let a,b,c,d> and <x,y< ,then

 +a cx

 +_dby<

F(a;c;x)

F(b;d;y)

< +

a cx

 +b_dy. (.)

(ii)Let c>a> ,d>b> ,x> and <y< ,then

 –a_cx

 –_dby+_b_d(₍b_d+)_+)y <

F(a;c; –x)

F(b;d; –y)

< –

a cx+

a(a+) c(c+)x

 –_dby . (.)

In general, a new generalized form of the conﬂuent hypergeometric functionFis

in-troduced by using the integral representation method

F(a,b;c,d;x,y) =

h,k≥

(a)h(b)k

h!k!(c)h(d)k

xhyk, c,d> ,x,y> . (.)

Now, let us give a generalization for the above inequalities.

Theorem . (i)Let a,b,c,d> and <x,y< ,then

 +a

cx+ b

dy<F(a,b;c,d;x,y) <  +

a c x+

b

dy. (.)

(ii)Let a,a,b,b,c,c,d,d> and <x,y,z,w< ,then

 +a_cx+_dby

 +_caz+_dbw

< F(a,b;c,d;x,y)

F(a,b;c,d;z,w)

< +

a cx+

b dy

 +a_cz+b_dw

. (.)

(iii)Let c>a> ,d>b> ,c>a> ,d>b> ,x,y> ,and <z,w< ,then

 –a_cx–b_dy

 –_caz–b_dw+_a_c(₍a_c+)_+)z+_ca_dbzw+_b_d(₍b_d+)_+)w

< F(a,b;c,d; –x, –y)

F(a,b;c,d; –z, –w)

< –

a cx–

b dy+

a(a+) c(c+)x

₊ab cdxy+

b(b+) d(d+)y



 –a_cz–b_dw

. (.)

2 Inequalities for simple Laguerre functions

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The simple Laguerre functionLa(x) of one variable is deﬁned by []

La(x) =F(–a; ;x). (.)

In the following some inequalities are established.

Theorem . If a< and <x< ,then we have

 –ax<La(x) <  – ax. (.)

Proof Equation (.) follows directly from (.) and (.) by puttingc= .

Theorem . (i)Let– <a< and y>x> ,then

ex–y<La(x)

La(y)

< . (.)

(ii)If a,b< and <x,y< ,then

 –ax

 – by< La(x)

Lb(y)

< – ax

 –by . (.)

(iii)Let– <a,b< ,x> ,and <y< ,then

 +ax

 +by+b(b_–)y < La(–x)

Lb(–y)

< +ax+

a(a–)  x

 +by . (.)

Proof Similarly result (.), (.), and (.) can also be obtained with the help of (.), (.),

(.), and (.).

Our interest is to show that our inequality (.) gives inequality atx= . anda= –.;

. <L–.(.) < ..

Note also that not only the restriction that the argument is less than the order is waived, but also it holds in the extended domaina∈(–, ),x= ., andy= 

. < L–.(.)

L–.()

< .

As the examples fora= –., –.,b= –., –.,x= ., andy= ., we have from (.)

.

< L–.(.)

L–.(.)

< .,

. <L–.(.)

L–.(.)

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Fora= –., –.,b= –., –.,x= ., andy= . from (.), we have

.

< L–.(–.)

L–.(–.)

< .,

.

<L–.(–.)

L–.(–.)

< ..

Note also that in the common domain of validity, whereas (.) gives an improved lower and upper bounds, (.) gives only an improved lower bound.

Next, we deﬁne the simple Laguerre function of two variables by

La,b(x,y) =F(–a,b; , ;x,y). (.)

Similar investigations are developed to establish new inequalities for a simple Laguerre function of two variables.

Theorem . (i)Let a,b< and <x,y< ,then

 –ax–by<La,b(x,y) <  – ax– by. (.)

(ii)If a,b,c,d< and <x,y,z,w< ,then

 –ax–by

 – cz– dw<

La,b(x,y)

Lc,d(z,w)

< – ax– by

 –cz–dw . (.)

(iii)Let– <a,b,c,d< ,x,y> ,and <z,w< ,then

 +ax+by

 +cz+dw+c(c_–)z₊_cdzw₊d(d–)  w

< La,b(–x, –y)

Lc,d(–z, –w)

< +ax+by+

a(a–)  x

₊_abxy₊b(b–)  y



 +cz+dw . (.)

Proof From (.), (.), (.), (.), and (.), we obtain the inequalities (.), (.), and

(.).

The numerical computation appended below veriﬁes these ratios under suitable restric-tions, and gives bounds of ratios otherwise readily available. For the examples

. <L–.,–.(., .) < .,

. <L–.,–.(., .)

L–.,–.(., .)

< .,

. < L–.,–.(–., –.)

L–.,–.(–., –.)

< ..

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3 Inequalities for classical Laguerre functions

The classical Laguerre functionL(aα)(x) of one variable of orderαis deﬁned as

L(_aα)(x) = ( +α)a

(a+ )F(–a;  +α;x). (.)

Proceeding in a similar manner we can derive bounds for classical Laguerre function of one variable, but details are omitted.

Theorem . For a< ,α> –,and <x< ,we have

 – a α+ x

( +α)a

(a+ )

<L(_aα)(x) <

 – a α+ x

( +α)a

(a+ )

. (.)

Proof This can be proved by using (.) and (.).

Theorem . (i)Ifα+  > –a> and y>x> ,then

ex–y<L

(α)

a (x)

L(aα)(y)

< . (.)

(ii)Ifβ+  >α+  > –a> ,then

L(aα)(x)

L(aβ)(x)

> . (.)

(iii)If a,b< and <x,y< ,then

 –_αa_+x

 –_β_+by

(b+ )( +α)a

(a+ )( +β)b

<L

(α)

a (x)

L(_bβ)(y)<

 –_α_+ax

 –_βb_+y

(b+ )( +α)a

(a+ )( +β)b

. (.)

(iv)Ifα+  > –a> ,β+  > –b> ,x> and <y< ,then

_{ +} a

α+x

 +_βb_+y+_(_βb_+)((b–)_β_+)y

(b+ )( +α)a

(a+ )( +β)b

<L

(α)

a (–x)

L(_bβ)(–y)<

_{ +} a

α+x+

a(a–) (α+)(α+)x

 +_βb_+y

(b+ )( +α)a

(a+ )( +β)b

. (.)

Proof The proof of the theorem is similar to the proof of Theorem ..

For the set of valuesx= . andy= , we have from the inequalities (.) and (.)

. <L()_–.(.) < .,

.

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Some special cases of inequalities for classical Laguerre function are listed below:

.

<L

(.) –.(.)

L(.)_–.(.)< .,

.

<L

(.) –.(.)

L(.)_–.(.)< .,

. <L

(.) –.(.)

L(.)_–.(.)< .,

and

. <L

(.) –.(.)

L(.)_–.(.)< ..

The previous properties can be generalized as follows: We deﬁne the classical Laguerre functionL(_aα_,_b,β)(x,y) of two variables in the form

L(_aα_,_b,β)(x,y) = ( +α)a( +β)b

(a+ )(b+ )F(–a, –b;  +α,  +β;x,y). (.)

Theorem . (i)Let a,b< ,α,β> –,and <x,y< ,then

 – a  +αx–

b

 +βy

( +α)a( +β)b

(a+ )(b+ )

<L(_aα_,_b,β)(x,y) <

 – a  +αx–

b

 +βy

( +α)a( +β)b

(a+ )(b+ )

. (.)

(ii)Let a,b,c,d< and <x,y,z,w< ,then

_{ –} a

+αx–

b

+βy

 –_μ_+c z–_ν_+dw

(c+ )(d+ )( +α)a( +β)b

(a+ )(b+ )( +μ)c( +ν)d

< L

(α,β)

a,b (x,y)

L(_cμ_,_d,ν)(z,w)<

_{ –} a

α+x– b

+βy

 –_+c_μz–_+d_νw

(c+ )(d+ )( +α)a( +β)b

(a+ )(b+ )( +μ)c( +ν)d

. (.)

(iii)Letα+  > –a> ,β+  > –b> ,μ+  > –c> ,ν+  > –d> ,x,y> ,and <z,w< ,

then

_{ +} a

α+x+

b

β+y

 + c

μ+z+

d

ν+w+

c(c–) (μ+)(μ+)z+

cd

(μ+)(ν+)zw+

d(d–) (ν+)(ν+)w

×

(c+ )(d+ )( +α)a( +β)b

(a+ )(b+ )( +μ)c( +ν)d

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< L

(α,β)

a,b (–x, –y)

L(_cμ_,_d,ν)(–z, –w)<

_{ +} a

α+x+

b

β+y+

a(a–) (α+)(α+)x+

ab

(α+)(β+)xy+

b(b–) (β+)(β+)y

 +_μc_+z+_νd_+w

×

(c+ )(d+ )( +α)a( +β)b

(a+ )(b+ )( +μ)c( +ν)d

. (.)

Proof The proof of the theorem is very similar to Theorem ..

In particular, for the set of values α = , ., ., β= , ., ., x= ., ., and y= ., ., we have from (.), (.), and (.), respectively, the inequalities:

. <L(,)_{–.,–.}(., .) < ., . <L(.,.)_{–.,–.}(., .) < .,

. <L

(.,.)

–.,–.(., .)

L(.,.)_{–.,–.}(., .)< ..

In the next section, we discuss some inequalities for pseudo-Laguerre functions for dif-ferent values that exhibit very interesting behavior of these function.

4 Inequalities for pseudo-Laguerre functions

For the pseudo-Laguerre function of one variablefa(x;λ) [, ] deﬁned by

fa(x;λ) =

(–λ)a

(a+ )F(–a;  +λ–a;x), (.)

whereλis non-integral, the following inequalities for pseudo-Laguerre function are es-tablished.

Theorem . For the pseudo-Laguerre functions fa(x;λ)we have the inequalities:

 – a  +λ–ax

(–λ)a

(a+ )

<fa(x;λ) <

 – a  +λ–ax

(–λ)a

(a+ )

;

a< ,  <x< , (.)

ex–y<fa(x;λ)

fa(y;λ)

< ;  +λ–a> –a> ,y>x> , (.)

fa(x;λ)

fa(y;μ)

> (–λ)a (–μ)a

( +λ–a)( +μ)

( +μ–a)( +λ);  +μ–a>  +λ–a> –a> ,y>x> , (.)

 –_+_λa_–_ax

 –_+_μb_–_by

(b+ )(–λ)a

(a+ )(–μ)b

< fa(x;λ)

fb(y;μ)

<

 –_+_λa_–_ax

 –_+_μb_–_by

(b+ )(–λ)a

(a+ )(–μ)b

;

a,b< ,  <x,y<  (.)

and

 +_λ_–a_a_+x

 +_+_μb_–_by+_(_μ_–_bb_+)((b–)_μ_–_b_+)y

(b+ )(–λ)a

(a+ )(–μ)b

<fa(–x;λ)

fb(–y;μ)

<

_{ +} a

λ–a+x+

a(a–) (λ–a+)(λ–a+)x



 + b

+μ–by

(b+ )(–λ)a

(a+ )(–μ)b

;

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Proof From (.)-(.) and (.), we obtain the inequalities for pseudo-Laguerre function

(.)-(.).

From (.) whenx= ., .,  andy= ., ., , we get

. <fa(.;λ)

fa(.;λ)

< ,

. < fa(.;λ)

fa(.;λ)

< ,

. < fa(;λ)

fa(;λ)

< .

Letλandμbe non-integral, we deﬁne the pseudo-Laguerre functionfa,b(x,y;λ,μ) of

two variables in the form

fa,b(x,y;λ,μ) =

(–λ)a(–μ)b

(a+ )(b+ )F(–a, –b;  +λ–a,  +μ–b;x,y). (.)

Next, we discuss further the pseudo-Laguerre function of two variables and the inequal-ities and properties investigated.

Theorem . (i)If a,b< and <x,y< ,then

 – a  +λ–ax–

b

 +μ–by

(–λ)a(–μ)b

(a+ )(b+ )

<fa,b(x,y;λ,μ) <

 – a  +λ–ax–

b

 +μ–by

(–λ)a(–μ)b

(a+ )(b+ )

. (.)

(ii)If a,b,c,d< and <x,y,z,w< ,then

_{ –} a

+λ–ax– b

+μ–by

 –_+_νc_–_cz–_+_ξd_–_dw

(c+ )(d+ )(–λ)a(–μ)b

(a+ )(b+ )(–ν)c(–ξ)d

<fa,b(x,y;λ,μ)

fc,d(z,w;ν,ξ)

<

_{ –} a

+λ–ax–

b

+μ–by

 –_+_νc_–_cz–_+_ξd_–_dw

(c+ )(d+ )(–λ)a(–μ)b

(a+ )(b+ )(–ν)c(–ξ)d

. (.)

(iii)Ifλ–a+  > –a> ,  +μ–b> –b> ,  +ν–c> –c> ,  +ξ–d> –d> ,x,y> ,

and <z,w< ,then

_{ +} a

λ–a+x+

b

μ–b+y

 +_+_νc_–_cz+_+_ξd_–_dw+_(_ν_–_cc_+)((c–)_ν_–_c_+)z₊ cd

(ν–c+)(ξ–d+)zw+

d(d–) (ξ–d+)(ξ–d+)w

×

(c+ )(d+ )(–λ)a(–μ)b

(a+ )(b+ )(–ν)c(–ξ)d

<fa,b(–x, –y;λ,μ)

fc,d(–z, –w;ν,ξ)

<

_{ +} a

λ–a+x+

b

μ–b+y+

a(a–) (λ–a+)(λ–a+)x+

ab

(λ–a+)(μ–b+)xy+

b(b–) (μ–b+)(μ–b+)y

 +_+c_ν_–_cz+_+_ξd_–_dw

×

(c+ )(d+ )(–λ)a(–μ)b

(a+ )(b+ )(–ν)c(–ξ)d

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Proof By using (.)-(.) and (.), we obtain the inequalities (.)-(.).

Numerically, Theorem . gives

. <f–.,–.(., .; ., .) < .,

. <f–.,–.(., .; ., .) < ..

In the next section, we show that there exist operational relations between conﬂuent hypergeometric functions and the various types of Shively’s pseudo-Laguerre functions.

5 Inequalities for Shively’s pseudo-Laguerre functions Shively [] studied the pseudo-Laguerre function of one variable by

Ra(α,x) =

(α)a

(a+ )(α)a

F(–a;α+a;x). (.)

By using inequalities of a conﬂuent hypergeometric function, we can obtain some in-equalities for Shively’s pseudo-Laguerre function in the following theorem.

Theorem .

 – a α+ax

(α)a

(a+ )(α)a

<Ra(α,x) <

 – a α+ax

(α)a

(a+ )(α)a

;

a< ,α+a> ,  <x< , (.)

ex–y<Ra(α,x)

Ra(α,y)

< ; α+a> –a> ,y>x> , (.)

Ra(α,x)

Ra(β,y)

>(α)a(β)a (α)a(β)a

(α+a)(β+ a)

(β+a)(α+ a); β+a>α+a> –a> , (.)

 –_αa₊_ax

 –_β₊b_by

(b+ )(α)a(β)b

(a+ )(α)a(β)b

<Ra(α,x)

Rb(β,y)

<

 –_α₊a_ax

 –_βb₊_by

(b+ )(α)a(β)b

(a+ )(α)a(β)b

;

a,b< ,α+a,β+b> ,  <x,y<  (.)

and

 +_αa₊_ax

 +_βb₊_by+_(_β₊b_b(₎₍b–)_β₊_b_+)y

(b+ )(α)a(β)b

(a+ )(α)a(β)b

<Ra(α, –x)

Rb(β, –y)

<

_{ +} a

α+ax+

a(a–) (α+a)(α+a+)x



 +_βb₊_by

(b+ )(α)a(β)b

(a+ )(α)a(β)b

;

α+a> –a> ,β+b> –b> ,x> ,  <y< . (.)

Proof This can be proved by using (.)-(.) and (.).

In particular, fora= –.,α= ,  andx= ., we have from (.) the inequalities

. <R–.(, .) < .,

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Next, we deﬁne the Shively’s pseudo-Laguerre function of two variables as

Ra,b(α,β,x,y) =

(α)a(β)b

(a+ )(b+ )(α)a(β)b

F(–a, –b;α+a,β+b;x,y). (.)

A generalization for the above inequalities is given as follows.

Theorem . (i)If a,b< ,α+ a> ,β+ b> ,and <x,y< ,then

 – a α+ax–

b

β+by

(α)a(β)b

(a+ )(b+ )(α)a(β)b

<Ra,b(α,β,x,y) <

 – a α+ax–

b

β+by

(α)a(β)b

(a+ )(b+ )(α)a(β)b

. (.)

(ii)If a,b,c,d< ,α+ a,β+ b,μ+ c,ν+ d> ,and <x,y,z,w< ,then

_{ –} a

α+ax– b

β+by

 –_μ₊c_cz–_ν₊d_dw

(c+ )(d+ )(α)a(β)b(μ)c(ν)d

(a+ )(b+ )(α)a(β)b(μ)c(ν)d

< Ra,b(α,β,x,y)

Rc,d(μ,ν,z,w)

<

_{ –} a

α+ax–

b

β+by

 –_μc₊_cz–_ν₊d_dw

(c+ )(d+ )(α)a(β)b(μ)c(ν)d

(a+ )(b+ )(α)a(β)b(μ)c(ν)d

. (.)

(iii)Ifα+a> –a> ,β+b> –b> ,μ+c> –c> ,ν+d> –d> ,x,y> ,and <z,w< ,

then

_{ +} a

α+ax+ b

β+by

 +_μc₊_cz+_ν₊d_dw+_(_μ₊c_c(₎₍c–)_μ₊_c_+)z₊ cd

(μ+c)(ν+d)zw+

d(d–) (ν+d)(ν+d+)w

×

(c+ )(d+ )(α)a(β)b(μ)c(ν)d

(a+ )(b+ )(α)a(β)b(μ)c(ν)d

< Ra,b(α,β, –x, –y)

Rc,d(μ,ν, –z, –w)

<

_{ +} a

α+ax+ b

β+by+

a(a–) (α+a)(α+a+)x+

ab

(α+a)(β+a)xy+

b(b–) (β+b)(β+b+)y

 +_μc₊_cz+_νd₊_dw

×

(c+ )(d+ )(α)a(β)b(μ)c(ν)d

(a+ )(b+ )(α)a(β)b(μ)c(ν)d

. (.)

Proof Using the inequalities of conﬂuent hypergeometric function (.)-(.) and (.), we obtain the inequalities for Shively’s pseudo-Laguerre function.

As a numerical veriﬁcation, we have from Theorem .:

. <R–.,–.(, , ., .) < .,

. <R–.,–.(., ., ., .) < .,

. <R–.,–.(., ., ., .) < .,

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In the next section, we discuss further the various types of Hermite functions and the inequalities and properties investigated.

6 Inequalities for Hermite functions

The Hermite function of one variableHa(x) is deﬁned by

Ha(x) =

(–)a_(_a_{+ )}

(a+ ) F

–a; ;x



. (.)

The above arguments establish the inequalities for a Hermite function Ha(x) in the

following theorem.

Theorem . (i)If a< and <x_{< ,}_then

 – ax(–)

a_(_a_{+ )}

(a+ )

<Ha(x) <

 – ax(–)

a_(_a_{+ )}

(a+ )

. (.)

(ii)If–_<a< and y>x> ,then

ex–y<Ha(x)

Ha(y)

< . (.)

(iii)If a,b< and <x_,_y_{< ,}_then

 – ax

 – by

(–)a_(_a_{+ )}₍_b_{+ )}

(–)b₍_a_{+ )}_(_b_{+ )}

<Ha(x)

Hb(y)

<

 – ax

 – by

(–)a_(_a_{+ )}₍_b_{+ )}

(–)b₍_a_{+ )}_(_b_{+ )}

. (.)

(iv)If–_<a,b< and x_{> ,  <}_y_{< ,}_then

 + ax

 + by₊

b(b– )y

(–)a_(_a_{+ )}₍_b_{+ )}

(–)b₍_a_{+ )}_(_b_{+ )}

<Ha(x)

Hb(y)

<

 + ax₊

a(a– )x

 + by

(–)a_(_a_{+ )}₍_b_{+ )}

(–)b₍_a_{+ )}_(_b_{+ )}

. (.)

It has given the following form for the operational relations of this Hermite function of one variableHa+(x) []:

Ha+(x) =

(–)a__(_a_{+ )}

(a+ ) xF

–a; ;x



. (.)

Using inequalities of conﬂuent hypergeometric functions appropriately, we obtain some rather complicated estimates.

Theorem .

 – ax



(–)a__(_a_{+ )}_x

(a+ )

<Ha+(x) <

 – ax



(–)a__(_a_{+ )}_x

(a+ )

;

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ex–y

x y

<Ha+(x)

Ha+(y)

<

x y

, –

<a< ,y

_>_x_{> ,} _(.)

Ha(x)

Ha+(y)

> ( – a) (a+ )y, –



 <a< , (.)

 –_ax

 –_by

(–)a₍_b_{+ )}_(_a_{+ )}_x

(–)b₍_a_{+ )}_(_b_{+ )}_y

<Ha+(x)

Hb+(y)

<

 –_ax

 – by

(–)a₍_b_{+ )}_(_a_{+ )}_x

(–)b₍_a_{+ )}_(_b_{+ )}_y

;

a,b< ,  <x,y<  (.)

and

 +_ax

 +_by₊b(b–)  y

(–)a₍_b_{+ )}_(_a_{+ )}_x

(–)b₍_a_{+ )}_(_b_{+ )}_y

<Ha+(–x)

Hb+(–y)

<

_{ +} ax

₊a(a–)  x



 +_by

(–)a₍_b_{+ )}_(_a_{+ )}_x

(–)b₍_a_{+ )}_(_b_{+ )}_y

;

–

 <a,b< ,x

_{> ,  <}_y_{< .} _(.)

Proof From (.)-(.) and (.), it is easy to see that (.)-(.) hold.

For example, whena= –.,x= . andy= ., from (.) we get

. <H–.(.)

H–.(.)

< .

The previous results can be generalized as follows. We deﬁne the Hermite function

Ha,b(x,y) of two variables by the relation

Ha,b(x,y) =

(–)a+b_(_a_{+ )}_(_b_{+ )}

(a+ )(b+ ) F

–a, –b; ,

 ;x

_,_y

. (.)

A generalization for the above inequalities is given in the following.

Theorem . (i)Let a,b< and <x_,_y_{< ,}_then

 – ax– by(–)

a+b_(_a_{+ )}_(_b_{+ )}

(a+ )(b+ )

<Ha,b(x,y) <

 – ax– by(–)

a+b_(_a_{+ )}_(_b_{+ )}

(a+ )(b+ )

. (.)

(ii)Let a,b,c,d< and <x_,_y_,_z_,_w_{< ,}_then

 – ax_{– }_by

 – cz_{– }_dw

(–)a+b_(_a_{+ )}_(_b_{+ )}₍_c_{+ )}₍_d_{+ )}

(–)c+d₍_a_{+ )}₍_b_{+ )}_(_c_{+ )}_(_d_{+ )}

< Ha,b(x,y)

Hc,d(z,w)

<

 – ax_{– }_by

 – cz_{– }_dw

(–)a+b_(_a_{+ )}_(_b_{+ )}₍_c_{+ )}₍_d_{+ )}

(–)c+d₍_a_{+ )}₍_b_{+ )}_(_c_{+ )}_(_d_{+ )}

.

(13)

(iii)Let–_<a,b,c,d< ,x_,_y_{> ,}_and_{ <}_z_,_w_{< ,}_then

 + ax_{+ }_by

 + cz_{+ }_dw₊

c(c– )z+ cdzw+ 

d(d– )w

×

(–)a+b_(_a_{+ )}_(_b_{+ )}₍_c_{+ )}₍_d_{+ )}

(–)c+d₍_a_{+ )}₍_b_{+ )}_(_c_{+ )}_(_d_{+ )}

< Ha,b(x,y)

Hc,d(z,w)

<

 + ax+ by+_a(a– )x+ abxy+_b(b– )y

 + cz_{+ }_dw

×

(–)a+b_(_a_{+ )}_(_b_{+ )}₍_c_{+ )}₍_d_{+ )}

(–)c+d₍_a_{+ )}₍_b_{+ )}_(_c_{+ )}_(_d_{+ )}

. (.)

Proof From (.)-(.) and (.), it is easy to see that (.)-(.) hold.

Secondly, we deﬁne the Hermite functionHa+,b+(x,y) of two variables by the

conﬂu-ent hypergeometric functions as in the form

Ha+,b+(x,y) =

(–)a+b__(_a_{+ )}_(_b_{+ )}

(a+ )(b+ ) xyF

–a, –b; ,

 ;x

_,_y

. (.)

Theorem . (i)If a,b< and <x_,_y_{< ,}_then

 – ax _– by 

(–)a+b__(_a_{+ )}_(_b_{+ )}_xy

(a+ )(b+ )

<Ha+,b+(x,y) <

 – ax _– by 

(–)a+b(a+ )(b+ )xy

(a+ )(b+ )

. (.)

(ii)If a,b,c,d< and <x_,_y_,_z_,_w_{< ,}_then

 –_ax_– by

 –_cz_– dw

(–)a+b₍_c_{+ )}₍_d_{+ )}_(_a_{+ )}_(_b_{+ )}_xy

(–)c+d₍_a_{+ )}₍_b_{+ )}_(_c_{+ )}_(_d_{+ )}_zw

< Ha+,b+(x,y)

Hc+,d+(z,w)

<

 –_ax_– by

 – cz–

 dw

(–)a+b₍_c_{+ )}₍_d_{+ )}_(_a_{+ )}_(_b_{+ )}_xy

(–)c+d₍_a_{+ )}₍_b_{+ )}_(_c_{+ )}_(_d_{+ )}_zw

. (.)

(iii)If–_<a,b,c,d< ,x_,_y_{> }_and_{ <}_z_,_w_{< ,}_then

 +_ax+_by

 + cz+

 dw+

c(c–)  z+

cd

 zw+ d(d–)

 w

×

(–)a+b₍_c_{+ )}₍_d_{+ )}_(_a_{+ )}_(_b_{+ )}_xy

(–)c+d₍_a_{+ )}₍_b_{+ )}_(_c_{+ )}_(_d_{+ )}_zw

< Ha+,b+(–x, –y)

Hc+,d+(–z, –w)

<

 +_ax₊ by

₊a(a–)  x

₊ab

 x

_y₊b(b–)  y



 +_cz₊ dw

×

(–)a+b₍_c_{+ )}₍_d_{+ )}_(_a_{+ )}_(_b_{+ )}_xy

(–)c+d₍_a_{+ )}₍_b_{+ )}_(_c_{+ )}_(_d_{+ )}_zw

(14)

Proof Similarly the results (.)-(.) can also be obtained with the help of (.)-(.)

and (.).

In conclusion, Luke’s technique is to a large extent successful in obtaining inequalities for the confluent hypergeometric function. The theorems developed herein seem sufficient to indicate the general nature of expected results and we do not further pursue the sub-ject. Here, we have obtained inequalities for simple Laguerre, Laguerre, pseudo-Laguerre, Shively’s pseudo-Laguerre, and Hermite functions from a well-known result for conflu-ent hypergeometric functions and their confluconflu-ent cases. In a future work, we intend to investigate this aspect of the subject and to apply our techniques to develop inequalities for hypergeometric functions of several variables.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author expresses sincere appreciation to Dr. Mohamed Saleh Metwally (Department of Mathematics, Faculty of Science (Suez), Suez Canal University, Egypt), and Dr. Mahmoud Tawﬁk Mohamed (Department of Mathematics, Faculty of Science (New Valley), Assiut University, New Valley, EL-Kharga 72111, Egypt) for their kind interests, encouragement, help, suggestions, comments and the investigations for this series of papers. The author would like to thank the anonymous referees for valuable comments and suggestions, which have led to a better presentation of the paper.

Received: 1 October 2014 Accepted: 1 May 2015 References

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