INEQUALITIES ON THE LAMBERT W FUNCTION AND HYPERPOWER FUNCTION

(1)

LambertWFunction and Hyperpower Function

Abdolhossein Hoorfar and Mehdi Hassani vol. 9, iss. 2, art. 51, 2008

Title Page

Contents

JJ II

J I

Page1of 11 Go Back Full Screen

Close

INEQUALITIES ON THE LAMBERT W FUNCTION AND HYPERPOWER FUNCTION

ABDOLHOSSEIN HOORFAR MEHDI HASSANI

Department of Irrigation Engineering Department of Mathematics

College of Agriculture Institute for Advanced Studies in Basic Sciences

University of Tehran P.O. Box 45195-1159,

P.O. Box 31585-1518, Karaj, Iran Zanjan, Iran

EMail:hoorfar@ut.ac.ir EMail:mmhassany@member.ams.org

Received: 09 April, 2007

Accepted: 15 March, 2008

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 33E20, 26D07.

Key words: LambertW function, hyperpower function, special function, inequality.

Abstract: In this note, we obtain inequalities for the LambertW functionW(x), defined byW(x)e^W^(x) =xforx≥ −e⁻¹. Also, we get upper and lower bounds for the hyperpower functionh(x) =x^x^x.

..

.

(2)

Abdolhossein Hoorfar and Mehdi Hassani

vol. 9, iss. 2, art. 51, 2008

Title Page Contents

JJ II

J I

Close

1. Introduction

The LambertW functionW(x), is defined byW(x)e^W^(x) = xforx ≥ −e⁻¹. For

−e⁻¹ ≤x <0, there are two possible values ofW(x), which we take values not less than−1. The history of the function goes back to J. H. Lambert (1728-1777). One can find in [2] a more detailed definition ofW as a complex variable function, some historical background and various applications of it in Mathematics and Physics.

The expansion

W(x) = logx−log logx+

∞

X

k=0

∞

X

m=1

c_km(log logx)^m (logx)^k+m ,

holds true for large values of x, with ckm = ⁽⁻¹⁾_m!^kS[k +m, k + 1], where S[k + m, k+1]is Stirling cycle number [2]. The series in the above expansion is absolutely convergent and it can be rearranged into the form

W(x) =L₁−L₂+L₂

L₁ + L₂(L₂−2)

2L²₁ +L₂(2L²₂−9L₂+ 6) 6L³₁ +O

L₂ L₁

4! ,

whereL₁ = logx andL₂ = log logx. Note that bylog we mean logarithm in the basee. Since the Lambert W function appears in some problems in Mathematics, Physics and Engineering, it is very useful to have some explicit bounds for it. In [5]

it is shown that

(1.1) logx−log logx < W(x)<logx,

where the left hand side holds true forx > 41.19and the right hand side holds true forx > e. The aim of the present paper is to obtain some sharper bounds.

(4)

vol. 9, iss. 2, art. 51, 2008

Title Page Contents

JJ II

J I

Close

2. Some Sharp Bounds for the Lambert W Function

It is easy to see thatW(−e⁻¹) = −1, W(0) = 0andW(e) = 1. Also, forx > 0, sinceW(x)e^W^(x) =x > 0ande^W(x) >0, we haveW(x)>0. An easy calculation yields that

d

dxW(x) = W(x) x(1 +W(x)).

Thus,x_dx^dW(x)>0holds true forx >0and consequentlyW(x)is strictly increas- ing forx >0(and also for−e⁻¹ ≤x≤0, but not for this reason).

Theorem 2.1. For everyx≥e, we have

(2.1) logx−log logx≤W(x)≤logx− 1

2log logx,

with equality holding only forx = e. The coefficients−1and−¹₂ oflog logxboth are best possible for the rangex≥e.

Proof. For the given constant0< p≤2consider the function f(x) = logx− 1

plog logx−W(x), forx≥e. Obviously,

d

dxf(x) = plogx−1−W(x) px(1 +W(x)) logx, and ifp= 2, then

d

dxf(x) = (logx−W(x)) + (logx−1) 2x(1 +W(x)) logx .

(5)

vol. 9, iss. 2, art. 51, 2008

Title Page Contents

JJ II

J I

Close

Considering the right hand side of (1.1), we have _dx^df(x)> 0forx > eand conse- quentlyf(x)> f(e) = 0, and this gives right hand side of (2.1). Trivially, equality only holds forx=e. If0< p <2, then _dx^df(e) = ^p−2_2ep <0, and this yields that the coefficient−¹₂ oflog logxin the right hand side of (2.1) is the best possible for the rangex≥e.

For the other side, note thatlogW(x) = logx−W(x)and the inequalitylogW(x)

≤ log logx holds forx ≥ e, because of the right hand side of (1.1). Thus,logx− W(x) ≤ log logxholds forx ≥ e with equality only forx = e. The sharpness of (2.1) with coefficient−1forlog logxcomes from the relation lim

x→∞(W(x)−logx+ log logx) = 0. This completes the proof.

Now, we try to obtain some upper bounds for the functionW(x)with the main termlogx−log logx. To do this, we need the following lemma.

Lemma 2.2. For everyt∈Randy >0, we have (t−logy)e^t+y ≥e^t, with equality fort= logy.

Proof. Lettingf(t) = (t−logy)e^t+y−e^t, we have _dt^df(t) = (t−logy)e^t and

d²

dt²f(t) = (t+ 1−logy)e^t. Now, we observe thatf(logy) = _dt^df(logy) = 0and

d²

dt²f(logy) = y > 0. These show that the function f(t) takes its only minimum value (equal to0) att= logy, which yields the result of Lemma2.2.

Theorem 2.3. Fory > ¹_e andx >−¹_e we have

(2.2) W(x)≤log

x+y 1 + logy

, with equality only forx=ylogy.

(6)

vol. 9, iss. 2, art. 51, 2008

Title Page Contents

JJ II

J I

Close

Proof. Using the result of Lemma2.2witht=W(x), we get (W(x)−logy)e^W^(x)−(e^W(x)−y)≥0,

which, considering W(x)e^W(x) = x, gives (1 + logy)e^W^(x) ≤ x +y and this is desired inequality fory > ¹_e andx > −¹_e. The equality holds whenW(x) = logy, i.e.,x=ylogy.

Corollary 2.4. Forx≥ewe have

(2.3) logx−log logx≤W(x)≤logx−log logx+ log(1 +e⁻¹),

where equality holds in the left hand side forx = eand in the right hand side for x=e^e+1.

Proof. Consider (2.2) withy= ^x_e, and the left hand side of (2.1).

Remark 1. Taking y = x in (2.2) we getW(x) ≤ logx−log ^1+log₂ ^x

, which is sharper than the right hand side of (2.1).

Theorem 2.5. Forx >1,we have

(2.4) W(x)≥ logx

1 + logx(logx−log logx+ 1),

with equality only forx=e.

Proof. Fort >0andx >1, let

f(t) = t−logx

logx −(logt−log logx).

(7)

vol. 9, iss. 2, art. 51, 2008

Title Page Contents

JJ II

J I

Close

We have_dt^df(t) = _log¹_x−¹_t and_dt^d²2f(t) = _t¹2 >0. Now, we observe that_dt^df(logx) = 0and so

mint>0 f(t) = f(logx) = 0.

Thus, fort > 0 andx > 1we havef(t) ≥ 0, with equality att = logx. Putting t = W(x) and simplifying, we get the result, with equality at W(x) = logx, or, equivalently, atx=e.

Corollary 2.6. Forx >1we have

W(x)≤(logx)^1+log^log^x^x.

Proof. This refinement of the right hand side of (1.1) can be obtained by simplifying (2.4) withW(x) = logx−logW(x).

The bounds we have obtained up to now have the formW(x) = logx−log logx+

O(1). Now, we give bounds with the error termO(^{log log}_log_x^x)instead ofO(1).

Theorem 2.7. For everyx≥ewe have

(2.5) logx−log logx+1 2

log logx

logx ≤W(x)≤logx−log logx+ e e−1

log logx logx , with equality only forx=e.

Proof. Taking the logarithm of the right hand side of (2.1), we have

logW(x)≤log

logx−1

2log logx

= log logx+ log

1−log logx 2 logx

.

UsinglogW(x) = logx−W(x), we get

W(x)≥logx−log logx−log

1− log logx 2 logx

,

(8)

vol. 9, iss. 2, art. 51, 2008

Title Page Contents

JJ II

J I

Close

which, considering−log(1−t)≥tfor0≤t <1(see [1]) witht= ^{log log}_{2 log}_x^x, implies the left hand side of (2.5). To prove the other side, we take the logarithm of the left hand side of (2.1) to get

logW(x)≥log(logx−log logx) = log logx+ log

1− log logx logx

.

Again, usinglogW(x) = logx−W(x), we obtain

W(x)≤logx−log logx−log

1− log logx logx

.

Now we use the inequality −log(1−t) ≤ _1−t^t for0 ≤ t < 1 (see [1]) witht =

log logx

logx , to get

−log

1− log logx logx

≤ log logx logx

1−log logx logx

−1

≤ 1 m

log logx logx , where

m= min

x≥e

1− log logx logx

= 1− 1 e. Thus, we have

−log

1− log logx logx

≤ e e−1

log logx logx , which gives the desired bounds. This completes the proof.

(9)

vol. 9, iss. 2, art. 51, 2008

Title Page Contents

JJ II

J I

Close

3. Studying the Hyperpower Function h(x) = x

^x^x.

..

Consider the hyperpower functionh(x) =x^x^x.

..

. One can define this function as the limit of the sequence{h_n(x)}n∈Nwithh₁(x) =xandh_n+1(x) =x^hⁿ^(x). It is proven that this sequence converges if and only if e^−e ≤ x ≤ e¹^e (see [4] and references therein). This function satisfies the relation h(x) = x^h(x), which, on taking the logarithm of both sides and a simple calculation yields

h(x) = W(log(x⁻¹)) log(x⁻¹) .

In this section we obtain some explicit upper and lower bounds for this function.

Since the obtained bounds for W(x)hold for large values of x and since for such values ofxthe value oflog(x⁻¹)is negative, we cannot use these bounds to approx- imateh(x).

Theorem 3.1. Takingλ=e−1−log(e−1) = 1.176956974. . ., fore^−e ≤x≤e¹^e we have

(3.1) 1 + log(1−logx)

1−2 logx ≤h(x)≤ λ+ log(1−logx) 1−2 logx ,

where equality holds in the left hand side forx = 1 and in the right hand side for x=e¹^e.

Proof. For t > 0, we havet ≥ logt+ 1, which takingt = z −logz withz > 0, implies

z

1−2 log(z¹^z)

≥log

1−log(z¹^z) + 1,

(10)

vol. 9, iss. 2, art. 51, 2008

Title Page Contents

JJ II

J I

Close

and putting z¹^z = x, or equivalently z = h(x), yields that h(x)(1 −2 logx) ≥ log(1−logx) + 1; this is the left hand side (3.1), since1−2 logx is positive for e^−e≤x≤e¹^e. Note that equality holds fort=z =x= 1.

For the right hand side, we define f(z) = z − logz with ¹_e ≤ z ≤ e. We immediately see that 1 ≤ f(z) ≤ e− 1; in fact it takes its minimum value 1 at z = 1. Also, consider the function g(t) = logt−t+λ for 1 ≤ t ≤ e−1, with λ = e−1−log(e−1). Since _dt^dg(t) = ¹_t −1and g(e−1) = 0, we obtain the inequality logt −t+λ ≥ 0 for 1 ≤ t ≤ e−1, and putting t = z −logz with

1

e ≤z ≤ein this inequality, we obtain

log(1−logz) +λ ≥z

1−2 log(z¹^z) .

Takingz¹^z =x, or equivalentlyz = h(x)yields the right hand side (3.1). Note that equality holds forx=e¹^e (z =e, t =e−1). This completes the proof.

(11)

vol. 9, iss. 2, art. 51, 2008

Title Page Contents

JJ II

J I

Close

References

[1] M. ABRAMOWITZ AND I.A. STEGUN, Handbook of Mathematical Func- tions: with Formulas, Graphs, and Mathematical Tables, Dover Publications, 1972. [ONLINE: http://www.convertit.com/Go/ConvertIt/

Reference/AMS55.asp].

[2] R.M. CORLESS, G.H. GONNET, D.E.G. HARE, D.J. JEFFREY AND D.E.

KNUTH, On the Lambert W function, Adv. Comput. Math., 5(4) (1996), 329–

359.

[3] R.M. CORLESS, D.J. JEFFREYAND D.E. KNUTH, A sequence of series for the LambertW function, Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Kihei, HI), 197-204 (electronic), ACM, New York, 1997.

[4] I. GALIDAKIS AND E.W. WEISSTEIN, "Power Tower." From MathWorld–A Wolfram Web Resource. [ONLINE:http://mathworld.wolfram.com/

PowerTower.html].

[5] M. HASSANI, Approximation of the Lambert W function, RGMIA Research Report Collection, 8(4) (2005), Art. 12.

[6] E.W. WEISSTEIN, Lambert W-Function, from MathWorld–A Wolfram Web Resource. [ONLINE: http://mathworld.wolfram.com/

LambertW-Function.html].