Optimal lower and upper bounds for the geometric convex combination of the error function

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

For ......, the error function ...... is defined as ...... In this paper, we answer the question: what are the greatest value p and the least value q, such that the double inequality ...... holds for all ...... (or ......) and ......? Here, ...... (......), ...... and ...... are the weighted power and the weighted geometric mean, respectively.

Li et al. Journal of Inequalities and Applications (2015) 2015:382 DOI http://dx.doi.org/10.1186/s13660-015-0906-y

Web End =10.1186/s13660-015-0906-y

R E S E A R C H Open Access

Optimal lower and upper bounds for the geometric convex combination of the error function

http://crossmark.crossref.org/dialog/?doi=10.1186/s13660-015-0906-y&domain=pdf

Web End = Yong-Min Li1, Wei-Feng Xia2*, Yu-Ming Chu3 and Xiao-Hui Zhang3

*Correspondence: [email protected]

2School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, China Full list of author information is available at the end of the article

Abstract

For x R, the error function erf(x) is dened as

erf(x) = 2

[integraldisplay]

x0 et2 dt.

In this paper, we answer the question: what are the greatest value p and the least value q, such that the double inequalityerf(Mp(x, y; )) G(erf(x), erf(y); ) erf(Mq(x, y; )) holds for all x, y 1 (or 0 < x, y < 1)

and (0, 1)? Here, Mr(x, y; ) = (xr + (1 )yr)1/r (r = 0), M0(x, y; ) = xy1 and

G(x, y; ) = xy1 are the weighted power and the weighted geometric mean, respectively.

MSC: Primary 33B20; secondary 26D15

Keywords: error function; power mean; functional inequalities

1 Introduction

For x R, the error function erf(x) is dened as

erf(x) =

[integraldisplay]

x et dt.

The most important properties of this function are collected, for example, in [, ]. In the recent past, the error function has been a topic of recurring interest, and a great number of results on this subject have been reported in the literature []. It might be surprising that the error function has application in the eld of heat conduction besides probability [, ].

In , Aumann [] introduced a generalized notion of convexity, the so-called MN-convexity, when M and N are mean values. A function f : [, ) [, ) is MN-convex if f (M(x, y)) N(f (x), f (y)) for x, y [, ). The usual convexity is the special case when

M and N both are arithmetic means. Furthermore, the applications of MN-convexity reveal a new world of beautiful inequalities which involve a broad range of functions from the elementary ones, such as sine and cosine function, to the special ones, such as the

2015 Li et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License

(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro

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indicate if changes were made.

Li et al. Journal of Inequalities and Applications (2015) 2015:382 Page 2 of 8

function, the Gaussian hypergeometric function, and the Bessel function. For the details as regards MN-convexity and its applications the reader is referred to [].

Let (, ), we dene A(x, y; ) = x + ( )y, G(x, y; ) = xy, H(x, y; ) =

xy

y+()x

and Mr(x, y; ) = (xr +()yr)/r (r = ), M(x, y; ) = xy. These are commonly known as weighted arithmetic mean, weighted geometric mean, weighted harmonic mean, and weighted power mean of two positive numbers x and y, respectively. Then it is well known that the inequalities

H(x, y; ) = M(x, y; ) < G(x, y; ) = M(x, y; ) < A(x, y; ) = M(x, y; )

hold for all (, ) and x, y > with x = y.By elementary computations, one has

lim

r Mr(x, y;

and

) = max(x, y).

In [], Alzer proved that c() = +()erf()erf

(/()) and c() = are the best possible factors such that the double inequality

c() erf H(x, y; )

[parenrightbig] A[parenleftbig]erf(x),erf(y);

[parenrightbig] c() erf H(x, y; )

[parenrightbig] (.)

holds for all x, y [, +) and (, /).

Inspired by (.), it is natural to ask: does the inequality erf(M(x, y)) N(erf(x), erf(y)) hold for other means M, N, such as geometric, harmonic or power means?

In [, ], the authors found the greatest values , and the least values , , such that the double inequalities

erf M(x, y; )

[parenrightbig] A[parenleftbig]erf(x),

and

M(x, y; )

[parenrightbig]

hold for all x, y (or < x, y < ) and (, ).

In the following we answer the question: what are the greatest value p and the least value q, such that the double inequality

erf

Mp(x, y; )

[parenrightbig] G[parenleftbig]erf(x),

erf(y);

) = min(x, y) (.)

lim

r+ Mr(x, y;

erf(y);

[parenrightbig]

erf

M(x, y; )

[parenrightbig]

erf

erf

M(x, y; )

Mq(x, y; )

[parenrightbig]

holds for all x, y (or < x, y < ) and (, )?

2 Lemmas

In this section we present two lemmas, which will be used in the proof of our main results.

[parenrightbig] H[parenleftbig]erf(x),

erf(y);

[parenrightbig]

[parenrightbig]

erf

Li et al. Journal of Inequalities and Applications (2015) 2015:382 Page 3 of 8

Lemma . Let r = , r =

e erf() = . . . . , and u(x) = log erf(x/r). Then the

following statements are true:() if r < r, then u(x) is strictly convex on [, +); () if r r < , then u(x) is strictly concave on (, ];

() if r > , then u(x) is strictly concave on (, +).

Proof Simple computations lead to

u (x) = ex/rx/r

r erf(x/r) (.)

and

u (x) = ex/rx/r

r erf(x/r)g(x), (.)

where

g(x) =

x/r + r[parenrightbig] erf

x/r[parenrightbig]

ex/rx/r. (.)

Then

g (x) = x/rg(x), (.)

g(x) =

r

erf

x/r[parenrightbig] ex/rx/r, (.)

and

g (x) =

r ex/rx/r[bracketleftbig](r )x/r + r[bracketrightbig]. (.)

We divide the proof into two cases.

Case . If r < , then (.), (.), and (.) lead to

g (x) < , (.) lim

x

+ g(x) > , lim

x+ g(x) = , (.)

lim

x

+ g(x) = , lim

x+ g(x) = , (.)

and

g() = ( r) erf()

e . (.)

Inequality (.) implies that g(x) is strictly decreasing on [, +).

It follows from the monotonicity of g(x) and (.) that there exists x (, +), such that g(x) is strictly increasing on [, x] and strictly decreasing on [x, +).

From the piecewise monotonicity of g(x) and (.) we clearly see that there exists x (, +), such that g(x) < for x (, x) and g(x) > for x (x, +).

Li et al. Journal of Inequalities and Applications (2015) 2015:382 Page 4 of 8

Case .. If r < r, then from (.) we know that g() > . This leads to g(x) > for x [, +). Therefore (.) leads to the conclusion that u(x) is strictly convex on [, +).

Case .. If r r < , then (.) implies that g() . This leads to g(x) for x (, ]. Therefore (.) leads to the conclusion that u(x) is strictly concave on (, ].

Case . If r > , then (.) and (.) imply that

g(x) < (.)

and

lim

x

+ g(x) = (.)

for x (, +).

It follows from (.), (.), and (.) that g(x) < . Therefore (.) leads to the conclusion that u(x) is strictly concave on (, +).

Lemma . The function h(x) = x +

xex [integraltext]

x et

dt is strictly increasing on (, +).

Proof Simple computations lead to

h (x) = h(x)

(

[integraltext]

x

et dt)

, (.)

where

h(x) = x

[parenleftbigg][integraldisplay]

x et dt[parenrightbigg] + [parenleftbig] x[parenrightbig]ex [integraldisplay]

x et dt xex,

lim

+ h(x) = , (.)

and

x

h (x) = [parenleftbigg][integraldisplay]

x et dt[parenrightbigg] + [parenleftbig]x + x[parenrightbig]ex [integraldisplay]

x et dt + xex > (.)

for x (, +).

Hence, h(x) is strictly increasing on (, +), as follows from (.), (.), and (.).

3 Main results

Theorem . Let (, ) and r =

e erf() = . . . . . Then the double inequality

erf

Mp(x, y; )

[parenrightbig] G[parenleftbig]erf(x),

erf(y);

[parenrightbig]

erf

Mq(x, y; )

[parenrightbig] (.)

holds for all x, y if and only if p = and q r.

Proof First of all, we prove that inequality (.) holds if p = and q r. It follows from (.) that the rst inequality in (.) is true if p = . Since the weighted power mean

Li et al. Journal of Inequalities and Applications (2015) 2015:382 Page 5 of 8

Mt(x, y; ) is strictly increasing with respect to t on R, thus we only need to prove that the second inequality in (.) is true if r q < .

If r q < , u(z) = log erf(z/q), then Lemma .() leads to

u(s) + ( )u(t) u[parenleftbig]s + (

)t

[parenrightbig] (.)

for (, ) and s, t (, ].Let s = xq, t = yq, and x, y . Then (.) leads to the second inequality in (.). Second, we prove that the second inequality in (.) implies q r.

Let x and y . Then the second inequality in (.) leads to

D(x, y) =: erf

Mq(x, y; )[parenrightbig] G[parenleftbig]erf(x),

erf(y);

[parenrightbig] . (.)

It follows from (.) that

D(y, y) =

xD(x, y)|x=y =

and

x D(x, y)|x=y =

( )yerf (y) [bracketleftbigg]q + [parenleftbigg]y +

yey

[integraltext]

et dt [parenrightbigg][bracketrightbigg]

. (.)

y

Therefore,

q lim

y

+

y yey

[integraltext]

y

et dt [parenrightbigg]

= r

follows from (.) and (.) together with Lemma ..

Finally, we prove that the rst inequality in (.) implies p = . We distinguish two cases.

Case I. p . Then for any xed y [, +) we have

lim

x+

erf

Mp(x, y; )

[parenrightbig] =

and

lim

x+ G[parenleftbig]erf(x),

erf(y);

[parenrightbig] =

erf(y) < ,

which contradicts the rst inequality in (.).

Case II. < p < . Let x , = /p and y +. Then the rst inequality in (.) leads to

E(x) =: erf(x) erf(x) . (.)

It follows from (.) that

lim

x+ E(x) = (.)

Li et al. Journal of Inequalities and Applications (2015) 2015:382 Page 6 of 8

and

E (x) =

ex[bracketleftbigg]erf(x)

e()x[bracketrightbigg]. (.)

Note that > , then

lim

x+

erf(x)

e()x[bracketrightbigg] = . (.)

It follows from (.) and (.) that there exists a suciently large [, +), such that E (x) > for x (, +). Hence E(x) is strictly increasing on [, +).

From the monotonicity of E(x) on [, +) and (.) we conclude that there exists [, +), such that E(x) < for x (, +), this contradicts (.).

Theorem . Let (, ), then the double inequality

erf M(x, y; )

[parenrightbig] G[parenleftbig]erf(x),

erf(y);

[parenrightbig]

M(x, y; )

[parenrightbig] (.)

holds for all < x, y < if and only if r and .

Proof First of all, we prove that (.) holds if r and .

If r, u(z) = log erf(z/), then Lemma .() leads to

u

s + ( )t[parenrightbig] u(s) + ( )u(t) (.)

for (, ), s, t > .

Let s = x, t = y, and < x, y < . Then (.) leads to the rst inequality in (.). If , u(z) = log erf(z/), then Lemma .() leads to

u(s) + ( )u(t) u[parenleftbig]s + (

)t

erf

[parenrightbig] (.)

for (, ), < s, t < .

Therefore, the second inequality in (.) follows from s = x, t = y, and < x, y < together with (.).

Second, we prove that the second inequality in (.) implies .

Let < x, y < . Then the second inequality in (.) leads to

J(x, y) =: erf

M(x, y; )

[parenrightbig] G[parenleftbig]erf(x),

erf(y);

[parenrightbig] . (.)

It follows from (.) that

J(y, y) =

xJ(x, y)|x=y =

and

x J(x, y)|x=y =

( )y erf (y) [bracketleftbigg]

+

y + yey

[integraltext]

y

et dt [parenrightbigg][bracketrightbigg]

. (.)

Li et al. Journal of Inequalities and Applications (2015) 2015:382 Page 7 of 8

Hence, from (.) and (.) together with Lemma . we know that

lim

y

+

y + yey[integraltext]

y

et dt [parenrightbigg][bracketrightbigg]

= .

Finally, we prove that the rst inequality in (.) implies r. Let y . Then the rst inequality in (.) leads to

L(x) =: G

erf(x), erf();

[parenrightbig] erf

M(x, ; )

[parenrightbig] (.)

for < x < .It follows from (.) that

L() = (.)

and

L (x) =

ex

erf() erf(x) x[parenleftbig]x +

/ex(x+)/[bracketrightbig]. (.)

Let

L(x) = log

erf() erf(x)[bracketrightbig] log

x[parenleftbig]x +

/ex(x+)/[bracketrightbig]. (.)

Then

lim

x

L(x) = , (.)

L (x) = ( )

erf (x) erf(x)

( )( )x(x + ) x +

x[parenleftbig]x +

/,

and

lim

x

e erf()[bracketrightbigg]. (.)

If > r, then from (.) we clearly see that there exists a small > , such that L (x) < for x ( , ). Therefore, L(x) is strictly decreasing on [ , ].

The monotonicity of L(x) on [, ] and (.) imply that there exists > , such that L(x) > for x ( , ).

Hence, (.) and (.) lead to L(x) being strictly increasing on [ , ]. It follows from the monotonicity of L(x) and (.) that there exists > , such that L(x) < for x ( , ), this contradicts (.).

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally to the writing of this paper. All authors read and approved the nal manuscript.

L (x) = ( )[bracketleftbigg]

Li et al. Journal of Inequalities and Applications (2015) 2015:382 Page 8 of 8

Author details

1School of Science, Huzhou Teachers College, Huzhou, 313000, China. 2School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, China. 3School of Mathematics and Computation Sciences, Hunan City University, Yiyang, 413000, China.

Acknowledgements

This research was supported by the Natural Science Foundation of China under Grants 61174076, 61374086, 11371125, and 11401191, and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004. The authors wish to thank the anonymous referees for their careful reading of the manuscript and their fruitful comments and suggestions.

Received: 11 February 2015 Accepted: 23 November 2015

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