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Approximation of the Lambert W Function and Hyperpower Function

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FUNCTION

ABDOLHOSSEIN HOORFAR AND MEHDI HASSANI

Abstract. In this note, we get some explicit approximations for the LambertW functionW(x), defined by W(x)eW(x) = xfor x ≥ −e−1. Also, we get upper and lower bounds for the hyperpower function

h(x) =xxx... .

1. Introduction

The Lambert W functionW(x), is defined byW(x)eW(x)=xfor x≥ −e−1. For −e−1 ≤x < 0, there

are two possible values of W(x), which we take such values that aren’t less than −1. The history of the

function goes back to J. H. Lambert (1728-1777). One can find in [2] more detailed definition of W as a

complex variable function, historical background and various applications of it in Mathematics and Physics.

Expansion

W(x) = logx−log logx+ ∞

X

k=0 ∞

X

m=1 ckm

(log logx)m

(logx)k+m,

holds true for large values ofx, withckm=

(−1)k

m! S[k+m, k+ 1] whereS[k+m, k+ 1] is Stirling cycle number [2]. The series in above expansion being to be absolutely convergent and it can be rearranged into the form

W(x) =L1−L2+ L2 L1

+L2(L2−2) 2L2

1

+L2(2L 2

2−9L2+ 6) 6L3

1

+O

L

2 L1

4!

,

where L1 = logx and L2 = log logx. Note that by log we mean logarithm in the base e. Since

Lam-bert W function appears in some problems in Mathematics, Physics and Engineering, having some explicit

approximations of it is very useful. In [5] it is shown that

(1.1) logx−log logx < W(x)<logx,

which the left hand side holds true forx >41.19 and the right hand side holds true forx > e. Aim of present

note is to get some better bounds.

2. Better Approximations of the Lambert W Function

It is easy to see thatW(−e−1) =−1,W(0) = 0 andW(e) = 1. Also, forx >0, sinceW(x)eW(x)=x >0

andeW(x)>0, we haveW(x)>0. About derivation, an easy calculation yields that

d

dxW(x) =

W(x) x(1 +W(x)).

2000Mathematics Subject Classification. 33E20, 26D07.

Key words and phrases. LambertW function, hyperpower function, special function, inequality.

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So, xdxd W(x)>0 holds true forx >0 and consequentlyW(x) is strictly increasing forx >0 (and also for

−e−1x0, but not by 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 for x=e. The coefficients −1 and −1

2 of log logxboth are best possible for the range x≥e.

Proof. For constant 0< p≤2 consider the function

f(x) = logx−1

plog logx−W(x),

forx≥e. Easily

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 .

Considering right hand side of (1.1) implies dxdf(x)>0 forx > eand consequentlyf(x)> f(e) = 0, and this

gives right hand side of (2.1). Trivially, equality holds for onlyx=e. If 0< p <2, then dxdf(e) =p2ep2 <0,

and this yields that the coefficient−1

2 of log logxin the right hand side of (2.1) is best possible for the range x≥e.

For the another side, note that logW(x) = logx−W(x) and the inequality logW(x)≤log logxholds for

x≥e, because of the right hand side of (1.1). Thus, logx−W(x)≤log logxholds forx≥ewith equality

only for x =e. Sharpness of (2.1) with coefficient −1 for log logx comes 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 main term logx−log logx. To do

this we need the following lemma.

Lemma 2.2. For everyt∈Randy >0, we have

(t−logy)et+y≥et,

with equality fort= logy.

Proof. Letting

f(t) = (t−logy)et+y−et,

we have

d

dtf(t) = (t−logy)e

t

and

d2

dt2f(t) = (t+ 1−logy)e

t.

Now, we observe that

f(logy) = d

(3)

and

d2

dt2f(logy) =y >0.

This means the function f(t) takes its minimum value equal to 0 att= logy, only. This gives the result of

lemma.

Theorem 2.3. Fory > 1e andx >−1

e we have

(2.2) W(x)≤log

x+y 1 + logy

,

with equality only for x=ylogy.

Proof. Using the result of above lemma witht=W(x), we get

(W(x)−logy)eW(x)−(eW(x)−y)≥0,

which consideringW(x)eW(x)=xgives (1 + logy)eW(x)x+y and this is desired inequality fory > 1

e and

x >−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 left hand side forx=eand in left hand side forx=ee+1.

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

Remark 2.5. Taking y =xin (2.2) we getW(x)≤logx−log1+log2 x, which is sharper than right hand

side of (2.1).

Theorem 2.6. Forx >1 we have

(2.4) W(x)≥ logx

1 + logx(logx−log logx+ 1),

with equality only for x=e.

Proof. Fort >0 andx >1, let

f(t) = t−logx

logx −(logt−log logx), We have

d dtf(t) =

1 logx−

1 t, and

d2 dt2f(t) =

1 t2 >0, Now, we observe that d

dtf(logx) = 0 and so

min

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

Thus, for t >0 andx >1 we havef(t)≥0 with equality att= logx. Puttingt=W(x) and simplifying,

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Corollary 2.7. Forx >1we have

W(x)≤(logx)1+loglogxx.

Proof. This refinement of the right hand side of (1.1), can be obtained simplifying (2.4) with W(x) =

logx−logW(x).

Bounds which we have obtained up to now have the formW(x) = logx−log logx+O(1). Now, we give

bounds with error term O(log loglogxx) insteadO(1), with explicit constants for error term.

Theorem 2.8. For everyx≥e we 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 for x=e.

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

logW(x)≤log

logx−1

2log logx

= log logx+ log

1−log logx

2 logx

.

Using logW(x) = logx−W(x), we get

W(x)≥logx−log logx−log

1−log logx

2 logx

,

which considering−log(1−t)≥tfor 0≤t <1 (see [1]) witht= log log2 logxx, implies the left hand side of (2.5).

To prove another side, we take logarithm from the left hand side of (2.1) to get

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

1−log logx

logx

.

Again, using logW(x) = logx−W(x), we get

W(x)≤logx−log logx−log

1−log logx

logx

.

Now we use the inequality−log(1−t)≤ t

1−t for 0≤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. So, we have−log

1−log logx

logx

≤ e

e−1 log logx

logx , which gives desired

bounds. This completes the proof.

3. Studying the hyperpower function h(x) =xxx...

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

. One can define this function as the limit of the sequence

{hn(x)}n∈N with h1(x) = xand hn+1(x) =xhn(x). It is proven that this sequence converge if and only if

e−exe1

e (see [4] and references therein). This function satisfies the relationh(x) =xh(x), which taking

logarithm from both sides and a simple calculation yields

h(x) = W(log(x −1))

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In this section we get some explicit upper and lower bounds for this function. To do this we won’t use the

bounds of LambertW function, cause of they holds true and are sharp forxlarge enough. Instead, we do

it directly.

Theorem 3.1. Takingλ=e−1−log(e−1) = 1.176956974· · ·, for e−exe1

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 left hand side forx= 1 and in the right hand side forx=e1e.

Proof. Fort >0 we havet≥logt+1, which takingt=z−logzwithz >0, impliesz 1−2 log(zz1)≥log 1−

log(z1z)+ 1, and puttingz 1

z =x, or equivalentlyz=h(x), it yields thath(x)(1−2 logx)≥log(1−logx) + 1;

this is the left hand side (3.1), cause of 1−2 logxis positive fore−e≤x≤e1e. Note that equality holds for

t=z=x= 1.

For the right hand side, we define f(z) =z−logzwith 1e ≤z≤e. Easily we see that 1≤f(z)≤e−1; in

fact it takes its minimum value 1 atz= 1. Also, consider the functiong(t) = logt−t+λfor 1≤t≤e−1,

withλ=e−1−log(e−1). Since dtdg(t) =1t−1 andg(e−1) = 0, we obtain the inequality logt−t+λ≥0

for 1≤t≤e−1, and puttingt=z−logz with 1e ≤z≤ein this inequality, we obtain log(1−logz) +λ≥

z 1−2 log(z1z). Takingz 1

z =x, or equivalentlyz=h(x) yields the right hand side (3.1). Note that equality

holds forx=e1e (z=e, t=e−1). This completes the proof.

References

[1] M. Abramowitz and I.A. Stegun,HANDBOOK OF MATHEMATICAL FUNCTIONS: with Formulas, Graphs, and Mthe-matical Tables, Dover Publications, 1972.

Available at: http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP

[2] R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, D.E. Knuth, On the LambertW function,Adv. Comput. Math.5 (1996), no. 4, 329-359.

[3] Robert M. Corless, David J. Jeffrey, Donald 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] Ioannis Galidakis and Eric W. Weisstein, ”Power Tower.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/PowerTower.html

[5] Mehdi Hassani, Approximation of the Lambert W Function,RGMIA Research Report Collection, 8(4), Article 12, 2005. [6] Eric W. Weisstein. ”Lambert W-Function.” From MathWorld–A Wolfram Web Resource.

http://mathworld.wolfram.com/LambertW-Function.html

Abdolhossein Hoorfar,

Department of Irrigation Engineering, College of Agriculture, University of Tehran, P.O. Box 31585-1518, Karaj, Iran

E-mail address:hoorfar@ut.ac.ir

Mehdi Hassani,

Department of Mathematics, Institute for Advanced Studies in Basic Sciences, P.O. Box 45195-1159, Zanjan, Iran

References

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