Z. angew. Math. Phys. 49 (1998) 137–1550044-2275/98/010137-20 $ 1.50+0.20/0c© 1998 Birkhauser Verlag, Basel

Zeitschrift fur angewandteMathematik und Physik ZAMP

Bounds on effective moduli by analytical continuation ofthe Stieltjes function expanded at zero and infinity

S. Tokarzewski and J. J. Telega

Abstract. By employing special continued fractions to two Stieltjes series with nonzero radiiof convergence the inequalities for two- and three-point Pade approximants have been derived.These inequalities constitute an extension of the relevant relations valid for one-point Pade ap-proximants [1,Cor.17.1] and two-point Pade ones [23, Th.1]. The Pade approximants inequalitiesachieved have been applied to the theory of inhomogeneous media for establishing of the newbounds on the effective transport coefficient of two-phase composite materials. As an examplethe low order estimations of the effective conductivity of a square array of cylinders have beencomputed and compared with the corresponding ones evaluated in [24].

Mathematics Subject Classification (1991). 41A21, 73B27.

Keywords. Pade approximants, Stieltjes function, composites, bounds, effective conductivities.

1. Introduction

The macroscopic modelling of inhomogeneous media and composites often requiresthe evaluation of the effective moduli. However their exact values are availableonly in specific cases; for instance in one-dimensional periodic homogenization.In the relevant literature, many papers were concerned with estimating of theeffective coefficients (such as dielectric constant, magnetic permeability, thermal orelectrical conductivity) of two-phase media. Assume that the macroscopic responseλe(x) (x = h − 1, h = λ1/λ2) of the two-phase composite is isotropic, where λ1and λ2 characterize the two materials, which the composite is made of. Theproblem arises, how to determine the best bounds on λe(x) with respect to thegiven information?

Wiener [28] derived optimal bounds on λe(h) with prescribed volume fractions.These bounds are known as the arithmetic and harmonic mean bounds. Forisotropic materials Hashin and Shtrikman [16] improved Wiener’s bounds usingvariational principles. Bergman [4,5,6] introduced a method for obtaining boundson λe(h) which does not rely on variational principles. Instead it exploits the prop-erties of the effective parameters as analytic functions of the components moduli.

138 S. Tokarzewski and J. J. Telega ZAMP

The method of Bergman was studied in more detail and applied to several physicalproblems by Milton [22, 23]. A rigorous justification of Bergman’s approach wasgiven by Golden and Papanicolaou [13]. Recently, an interesting continued fractiontechniques for evaluation of the bounds on λe(h) have been presented by Bergman[7] for three- and Clark and Milton [9] for two-dimensional systems. Both Milton[23] and Bergman [7] have incorporated into bounds the power expansion of λe(h)at h = 1 and only discrete values of λe(h1), λe(h2),... ,λe(hK).

The present paper incorporates into bounds on λe(x) the power expansionsof λe(x) at x = 0 and x = ∞. That particular case, important for theoreticaland experimental investigations, has been studied very recently in the contextsof: (i) the estimations of Stieltjes functions by Pade approximants, cf.[24, 26] and(ii) the bounds on the effective conductivity of regular composites, cf.[25]. Froma fixed set of input data it has been derived in [24− 26] the estimations of λe(x)represented by two-point Pade approximants only.

The main aim of this paper is to establish new three-points Pade bounds on theeffective transport coefficients λe(x) generated by a given number of coefficients ofthe power expansions of λe(x) at x = 0 and x =∞. Two-point Pade approximants[26] and three-point Pade ones complete the estimations of λe(x) from the top andbelow, obtainable from an available fixed input data (p coefficients of an expansionof λe(x) at x = 0 and k coefficients at x =∞).

This paper is organized as follows: In Section 2 we introduce the basic def-initions, notations and assumptions dealing with a Stieltjes function xf1(x) anddiagonal two- and three-point Pade approximants to xf1(x). In Section 3 we recallthe relevant results for one-point Pade approximants. In Section 4 we derive two-and three-point continued fraction representations for xf1(x). The inequalities fordiagonal two- and three-point Pade approximants have been derived in Section5. In Section 6 the upper and lower bounds on the Stieltjes function xf1(x) areestablished. Numerical examples illustrating the bounds obtained are presentedin Section 7. In Section 8 the general bounds on the effective moduli of two- phasemedia are derived. The effective conductivity of a square array of cylinders hasbeen dealt with in Section 9. The results achieved are summarized in Section 10.The basic recurrence equations for finding two- and three-point Pade approximantsare presented in the Appendix.

2. Preliminaries

Let us consider a Stieltjes function xf1(x) defined for −R < x < ∞ by means ofthe following Stieltjes-integral:

xf1(x) = x

1/R∫0

dγ1(u)1 + xu

, 0 ≤ u ≤ 1/R, (2.1)

Vol. 49 (1998) Stieltjes function expanded at zero and infinity 139

wherelim

x→−R+= xf1(x) ≥ U ≥ −∞.

The spectrum γ1(u) is a real, bounded and non-decreasing function, while U is aknown real number. Consider the power expansion of xf1(x) at x = 0:

xf1(x) =∞∑n=1

c(1)n xn, (2.2)

where the coefficients

c(1)n = (−1)n+1

1/R∫0

un−1dγ1(u), (2.3)

are real and finite. Note that power series (2.2) has the radius of convergence atleast R. The power expansion of xf1(x) at x =∞ takes the form

xf1(x) =∞∑n=0

C(1)n sn, s = 1/x. (2.4)

Here the moments

C(1)n = (−1)n

1/R∫0

u−1−ndγ1(u), n = 0, 1, 2, ... (2.5)

are assumed to be finite for any fixed n. Diagonal two-point [M/M ]k and three-point [M/M ]k Pade approximants for series (2.2) and (2.4) are defined by thefollowing rational functions:

[M/M ]k =a1kx+ a2kx

2 + ...+ aMkxM

1 + b1kx+ b2kx2 + ...+ bMkxM,

[M/M ]k =a1kx+ a2kx

2 + ...+ aMkxM

1 + b1kx+ b2kx2 + ...+ bMkxM.

(2.6)

Consider the power expansions of (2.6) at zero:

[M/M ]k =∞∑n=1

cnkxn , [M/M ]k =

∞∑n=1

cnkxn, (2.7)

140 S. Tokarzewski and J. J. Telega ZAMP

and at infinity

[M/M ]k =∞∑n=0

Cnksn , [M/M ]k =

∞∑n=0

Cnksn, s = 1/x. (2.8)

Now we may formulate

Definition 1. The rational function (2.6)1 is a two-point Pade approximant [M/M ]kto the Stieltjes function (2.1) if

cnk = c(1)n for n = 1, 2, . . . , p , p = 2M − k, (2.9)

andCnk = C

(1)n for n = 0, 1, . . . , k − 2 , C(k−1)k = C

(1)k−1 . (2.10)

We observe that according to the above definition [M/M ]0 stands for the one-point Pade approximant.

Definition 2. The rational function (2.6)2 is a three-point Pade approximant[M/M ]k to the Stieltjes function (2.1) if

cnk = c(1)n for n = 1, 2, . . . , p , p = 2M − k, (2.11)

and

Cnk = C(1)n for n = 0, 1, . . . , k − 2 , lim

x→−R+[M/M ]k = U, (2.12)

where R and U were introduced earlier, cf.(2.1).

The above definitions will be used for the construction of bounds on the Stielt-jes function xf1(x) from the power series (2.2) and (2.4) with non-zero radii ofconvergence. In our previous investigations only the two-point Pade approximantswere used, cf.[24 − 26]. Three-point Pade approximants lead to improvement ofthe bounds discussed in [24− 26]. Moreover, the two- and three point Pade esti-mations of xf1(x) appear to be the best with respect to the parameters U,R andgiven number of coefficients of the power series (2.2) and (2.4),cf.(2.1).

Throughout this paper the parameter p will denote a number of available coef-ficients of the power series (2.2), while k a number of relations given by (2.10) ifwe deal with [M/M ]k or by (2.12) if we study [M/M ]k, where p+ k = 2M .

Vol. 49 (1998) Stieltjes function expanded at zero and infinity 141

3. A brief account of the properties of one-point Pade approxi-mants [M/M ]0

We start our discussion by recalling some results for one-point Pade approximants[M/M ]o to xf1(x), indispensable for our further investigations. Those results maybe summarized as follows:1) [M/M ]0 has the continued fraction representation of the type S [1-3]

[M/M ]o =xg11 +

xg21 +...+

xg2M−11 +

xg2M1

(3.1)

2) The coefficients of the continued fraction (3.1) are positive

gn > 0. (3.2)

3) For x > −R Pade approximants [M/M ]o and [M + 1/M ]o to power series (2.2)converge monotonically and uniformly to the Stieltjes function (2.1) on compactsubsets of (−R,∞), cf.[1, Th.16.2]. We may write

limM→∞

[M + 1/M ]0 = limM→∞

[M/M ]0 = xf1(x) , x > −R. (3.3)

4) If xfj(x) (j = 1, 2, . . . ) is a Stieltjes function

xfj(x) = x

1/R∫0

dγj(u)1 + xu

, (3.4)

then the function xfj+1(x) given by

xfj+1(x) = x

1/R∫0

dγj+1(u)1 + xu

, (3.5)

is also a Stieltjes function, provided that

fj(x) =fj(0)

1 + xfj+1(x), x > 0, (3.6)

cf. [1, Lemma 15.3] and [1, p.235]. �If the expansion of xfj(x) at x =∞ is given by

xfj(x) =∞∑n=0

C(j)n (1/x)n, C(j)

0 > 0, (3.7)

142 S. Tokarzewski and J. J. Telega ZAMP

then on account of (3.6) the expansion of xfj+1(x), also at x =∞ takes the form

xfj+1(x) = C(j+1)x+∞∑n=0

Cn(1/x)n, C(j+1) =fj(0)C 0

> 0. (3.8)

Consequently we have the following relations

xfj+1(x) = C(j+1)x+ xfj+2(x), (3.9)

where

fj+2(x) =

1/R∫0

dγj+2(u)1 + xu

, (3.10)

is a Stieltjes function of the type (2.1).The relations (3.7)− (3.10) permit us to draw the following conclusion:If xfj(x) is a Stieltjes function:

xfj(x) = x

1/R∫0

dγj(u)1 + xu

=∞∑n=0

C(j)n sn, s = 1/x, (3.11)

then xfj+2(x) is also a Stieltjes function

xfj+2(x) = x

1/R∫0

dγj+2(u)1 + xu

, (3.12)

provided that

fj(x) =fj(0)

1 + C(j+1)x+ xfj+2(x), C(j+1) =

fj(0)

C(j)0

. � (3.13)

The fractional transformations (3.6) and (3.13) will be used for the construction ofspecial continued fractions representing two- and three-point Pade approximants[M/M ]k and [M/M ]k respectively. Additionally, without loss of generality, weassume p ≥ k, cf. Sec 2.

Vol. 49 (1998) Stieltjes function expanded at zero and infinity 143

4. Continued fraction representation for Pade approximants

Let us apply the fractional transformation (3.13) to xf1(x) k times. Thus weobtain T -continued fraction to xf1(x), cf. [14,19],

xf1(x) =xG1

1 + xG2 +xG3

1 + xG4 +...+xG2k−1

1 + xG2k + xf2k+1(x). (4.1)

Here the parameters Gn are uniquely determined by k coefficients of power series(2.2) and k terms of expansion (2.4). The function xf2k+1(x) appearing in (4.1)is the Stieltjes function of type (2.1). Now we can construct S-continued fractionto xf2k+1(x). By employing (3.6) to xf2k+1(x) (p − k) times we arrive at

xf2k+1(x) =xg2k+1

1 +xg2k+2

1 +...+xgp+k

1 + xfp+k+1(x), p ≥ k. (4.2)

The substitution of (4.2) into (4.1) leads to the following continued fraction rep-resentation for [M/M ]k, cf. Def.1,

[M/M ]k =xG1

1 + xG 2 +...+xG2k−1

1 + xG 2k+xg2k+1

1 +...+xgp+k

1, (4.3)

and for [M/M ]k, cf. Def.2,

[M/M ]k =xG1

1 + xG2 +...+xG2k−1

1 + xG2k+xg2k+1

1 +...+xgp+k

1 +xVp+k+1

1, (4.4)

where the parameter Vp+k+1 satisfies

1+xG2+. . .+xG2k−1

1 + xG2k+xg2k+1

1 +...+xgp+k

1 +xVp+k+1

1= 0 , if x = −R, (4.5)

whilep+ k = 2M. (4.6)

The coefficients Gn (n = 1, 2, . . . , 2k), g2k+j (j = 1, 2, . . . , p − k) and Vp+k+1appearing in (4.3)− (4.4) are positive, i.e.:

Gn > 0 , n = 1, 2, ..., 2k ; g2k+j > 0 , (j = 1, 2, ..., p−k) ; Vp+k+1 > 0. (4.7)

On account of [1, Th.16.2] we infer

limp→∞

xg2k+11 +

xg 2k+21 +...+

xgp+k1 + xfp+k+1(x)

= xf2k+1(x) , −R < x <∞.

(4.8)

144 S. Tokarzewski and J. J. Telega ZAMP

Hence the relation (4.1) leads to

limM→∞

[M/M ]k = xf1(x) , −R < x <∞. (4.9)

On the basis of the recurrence formula for the continued fraction (4.5) given inAppendix by (A.4) combined with relation (4.7) one can prove that

(−1)k[M/M ]k < (−1)k[M/M ]k < (−1)k[M + 1/M + 1]k , x > 0. (4.10)

Hence from (4.9), (4.10) and the fact that for | x |< R the power expansion of[M/M ]k at x = 0 converges to xf1(x), the following convergence readily follows

limM→∞

[M/M ]k = xf1(x) , −R < x <∞. (4.11)

Thus we conclude that for any fixed k (k = 0, 1..., 2M) two- and three- pointPade approximants [M/M ]k and [M/M ]k converge to a Stieltjes function xf1(x)for −R < x <∞, as M goes to infinity, cf. (4.9) and (4.11).

The Stieltjes function xfp+k+1(x) increases for x ∈ (−R,∞) and satisfies theinequalities valid for one-point Pade approximants, cf. [1,Th.15.2]. Hence oneobtains

xVp+k+1 < xfp+k+1(x) , if −R < x < 0,xVp+k+1 > xfp+k+1(x) , if x > 0,

(4.12)

xgp+k+1 > xfp+k+1(x) , if −R < x <∞. (4.13)

Here Vp+k+1 is determined by Eq.(4.5). Now we are prepared to investigate theanalytical properties of two- and three-point Pade approximants [M/M ]k and[M/M ]k to xf1(x) ensuing directly from the recurrence formulae for (4.3)− (4.4)given by (A.4), the constraints (4.7) and (4.12)−(4.13), and the convergence radiusR of the expansion (2.2).

5. Inequalities for [M/M]k and [M/M]k

We proceed to studying the basic inequalities for unbalanced two- and three-pointPade approximants [M/M ]k and [M/M ]k. To this end, for n = 1, 2, . . . , 2n− 1 <2k, we introduce a T - fraction operator:

T 2k2n−1(·) =

xG2n−11 + xG2n+

xG2n+11 + xG2n+2 +...+

xG2k−11 + xG2k+

(·)1, (5.1)

and for p = k, k + 1, . . . the following S- fraction operator:

Sp+k2k+1(·) =xg 2k+1

1 +xg2k+2

1 +...+xgp+k

1 +(·)1. (5.2)

Vol. 49 (1998) Stieltjes function expanded at zero and infinity 145

The Pade approximants (4.3)− (4.4) can then be written in the form:

[M/M ]k = T 2k1 Sp+k2k+1(0),

[M/M ]k = T 2k1 Sp+k2k+1(xVp+k+1),

(5.3)

where M and p, k are interrelated by (4.6). From Baker’s approach to the Stieltjesseries with nonzero radius of convergence [1,p.235], it follows that for x ≥ −R andY = 0 or Y = xVp+k+1 we have

1 + xG2 + T 2k3 Sp+k2k+1(Y ) ≥ 0, 1 + xG4 + T 2k

5 Sp+k2k+1(Y ) > 0,. . . , 1 + xG2k + Sp+k2k+1(Y ) > 0,

1 + Sp+k2k+2(Y ) > 0, 1 + Sp+k2k+3(Y ) > 0,. . . , 1 + Sk+p

k+p(Y ) > 0.

(5.4)

On account of (5.4) and (4.7), and from the recurrence formulae (A.4) one caneasily derive the basic inequalities for diagonal two- and three-point Pade approx-imants [M/M ]k and [M/M ]k to Stieltjes function xf1(x):(i) If −R < x < 0 then

[M/M ]k > [M/M ]k−1 > [M + 1/M + 1]k,

[M/M ]k < [M/M ]k−1 < [M + 1/M + 1]k.(5.5)

(ii) If 0 < x <∞ then

(−1)k[M + 1/M + 1]k > (−1)k[M/M ]k,

(−1)k[M + 1/M + 1]k < (−1)k[M/M ]k.(5.6)

For k = 0 relation (5.6)1 reduces to well known inequalities for one- point Pade ap-proximants [1.Th.15.2], while for k = 0, 1, 2 to the formulae reported in [24, (5.3)].For k = M the expression (5.6) coincides with relations originally reported byGonzalez-Vera and Njastad [14, Th. 2.3].

6. Bounds on xf1(x)

From (4.9), (4.11), (5.5) and (5.6) the following inequalities for Stieltjes functionxf1(x) are derived:(i) If −R < x ≤ 0 then

[M/M ]k > xf1(x) > [M/M ]k. (6.1)

146 S. Tokarzewski and J. J. Telega ZAMP

Figure 1.Monotone sequences of two- and three- point Pade approximants [M/M ]k and [M/M ]k (k =1, 2; M = 2, 3, 4) forming the upper and lower bounds converging to the Stieltjes function ln((1+1000x)/(1 + x))/D, D = ln 1000, −0.001 ≤ x ≤ 0.

(ii) If x ≥ 0 then

(−1k)[M/M ]k < (−1k) xf1(x) < (−1k)[M/M ]k. (6.2)

The above formulae provide the best upper and lower bounds on the Stieltjesfunction xf1(x) provided that the radius of convergence R, p coefficients of thepower series (2.2), k terms of the power expansion (2.4) and the value U cf. (2.1)are given.

7. Illustrative example

For an illustration of the inequalities (5.5), (5.6), (6.1) and (6.2) let us considerthe following Stieltjes function

xf1(x) = x

1/0.001∫1

du

1 + xu=

1ln 1000

· ln 1 + 1000x1 + x

. (7.1)

Hence the power expansions of (7.1) at x = 0

xf1(x) =1

ln 1000·∞∑n=1

(−1)n−1(1000n − 1)n

xn (7.2)

Vol. 49 (1998) Stieltjes function expanded at zero and infinity 147

Figure 2.Monotone sequences of two- and three- point Pade approximants [M/M ]k and [M/M ]k (k =1, 2; M = 2, 3, 4) forming the upper and lower bounds converging to the Stieltjes function ln((1+1000x)/(1 + x))/D, D = ln 1000, 0.0001 ≤ x ≤ 100.

and x =∞

xf1(x) = 1 +1

ln 1000

∞∑n=1

(−1)n(1− 0.001n)n

sn , s = 1/x (7.3)

are the data for the computation of Pade approximants [M/M ]k, [M/M ]k to Stielt-jes function (7.1). The radius of convergence of the series (2.2) is equal to R =0.001, while U = −∞, cf. (2.12). By using the recurrence formulae (A.1)-(A.4)reported in Appendix A we have evaluated the convergents of the continued frac-tions (4.3) − (4.4). Monotone sequences of the upper and lower bounds [M/M ]kand [M/M ]k (k = 1, M = 2, 3, 4; k = 2, M = 2, 3, 4) converging to the function(7.1) are depicted in Figs 1,2. Note that the numerical results presented in Figs 1,2confirm the predictions following from the inequalities (5.5), (5.6), (6.1) and (6.2).

8. Bounds on the effective parameters

In this Section we will apply the inequalities (6.1) and (6.2) to the theory ofinhomogeneous media for the calculation of the effective transport coefficients forcomposite materials consisting of two isotropic components.

Let us consider the effective conductivity Λeof a two-phase medium for the casewhere the conductivity coefficients λ1and λ2 of both components are real. The

148 S. Tokarzewski and J. J. Telega ZAMP

bulk conductivity Λe is defined by the linear relationship between the volume-averaged temperature gradient < 5T > and the volume-averaged heat flux < J >

< J >= Λe(x) <∇T > . (8.1)

By Φ and 1−Φ we denote the matrix and the inclusion volume fractions, respec-tively. In general Λe will be a second-rank symmetric tensor, even when λ1 andλ2 are both scalars. Our study will be focused upon one of the diagonal values ofΛe denoted by λe(λ1, λ2).

The analytic properties of the bulk conductivity coefficient λe(λ1, λ2) wereexamined by Bergman in [4]. He noticed that λe(λ1, λ2)/λ1 = λe(1, λ2/λ1) isan analytical function in the complex plane except on the negative part of thereal axis. Motivated by Bergman’s result Golden and Papanicolaou [12] rigorouslyproved that the effective conductivity λe(x) expressed by (λe(x)/λ1) − 1 is theStieltjes function of type (2.1):

xf1(x) =λe(x)λ 1

− 1 = x

1∫0

dγ1(u)1 + xu

, x = h− 1, h = λ2/λ1. (8.2)

For composite materials investigated here R = 1 and U = −1, cf. [4, 5]. Conse-quently the inequalities (6.1) − (6.2) are valid for the effective moduli expressedby λe(x)/λ1 − 1. On the basis of that conclusion we can formulate the followinggeneral theorem, which solves the problem of bounds on effective moduli λe(x)/λ1of two phase media generated by power expansions at x = 0 and x =∞.

Theorem 8.1. For any fixed k ∈ N two- and three-point Pade approximants[M/M ]k and [M/M ]k to the effective transport coefficient (λe(x)/λ1) − 1 repre-sented by power expansions at x = 0 and x =∞ obey the following inequalities:(i) If −1 < x < 0 then

[M/M ]k > [M/M ]k−1 > [M + 1/M + 1]k,

[M/M ]k < [M/M ]k−1 < [M + 1/M + 1]k,(8.3)

1 + [M/M ]k > λe(x)/λ1 > 1 + [M/M ]k. (8.4)

(ii) If x > 0 then

(−1)k[M + 1/M + 1]k > (−1)k[M/M ]k,

(−1)k[M + 1/M + 1]k < (−1)k[M/M ]k,(8.5)

(−1)k(1 + [M/M ]k) < (−1)kλe(x)/λ1 < (−1)k(1 + [M/M ]k), (8.6)

Vol. 49 (1998) Stieltjes function expanded at zero and infinity 149

Figure 3.Unit cell for a square array of cylinders

where λe(x)/λ1 stands for the limit as M goes to infinity of 1 + [M/M ]k or 1 +[M/M ]k in x ∈ (−1,∞).

Proof. It follows immediately from (5.5), (5.6), (6.1) and (6.2). �

The bounds on λe(x)/λ1 (−1 < x < ∞) given by relations (8.4) and (8.6) arenew. For the case k = 0 they coincide with estimations of λe(x)/λ1 reported byMilton in [23], while for R = 0 reduce to the inequalities derived in [26]. It isworth noting that the inequalities (8.4) and (8.6) also can be derived by applyingthe multi-point continued fraction method of Bergman [7] or Y, Y - transformationprocedures of Clark and Milton [9] adapted to three- dimensional systems. Itmeans, that the bounds (8.4) and (8.6) can be obtained as a limiting case ofmulti-point bounds of [23], which precede the bounds of [7] and [9].

9. Effective conductivity of densely packed cylinders

Now we are in position to apply Th. 8.1 for the evaluation of low order boundsgiven by [M/M ]kand [M/M ]kon the effective conductivity of a composite materialcontaining densely spaced, equally-sized cylinders. To this end we set: Φ = πρ2-volume fraction of inclusions, ρ- the radius of cylinders, λ1−, λ2- conductivities ofthe matrix and inclusions and x = (λ2/λ1)− 1- normalized physical properties ofthe cylinders. The bulk conductivity λe(x) with isotropic symmetry is defined by,cf. (8.1)

< J >= λe(x) <∇T > . (9.1)

The averaging < . > is performed over the unit square cell, see Fig.3. The tem-

150 S. Tokarzewski and J. J. Telega ZAMP

Figure 4.Low order bounds for the effective conductivity λe(x)/λ1 of a square array of cylinders: present[2/2]2, [2/2]2 and previous [2/2]1, [2/2]1 evaluated in [24]

perature field T appearing in (9.1) satisfies:(i) The conductivity equation

∇ · (1 + xθ)∇T = 0 , x = (λ2/λ1)− 1, (9.2)

where θ is the characteristic function of cylinders: θ = 1, if x belongs to thedomain occupied by cylinders, θ = 0 otherwise.(ii) The continuity condition

m · J− = m · J+, (9.3)

where m is the unit vector normal to the surface of a cylinder, while J− and J+denote the heat flux J = (1 + xθ)∇T on the inside and outside of the cylindersurface.

It is assumed that the normalized temperature gradient

<∇T >= 1 (9.4)

is directed along axis y1(Fig.3). Low order terms of the expansions of (λe(x)/λ)1−1at x = 0 and x = ∞, calculated from (9.1) − (9.4), are reported in [4] and [21]respectively, and have the following form

λe(x)/λ1 − 1 = Φx− 0.5Φ(1− Φ)x2 +O(x3) , Φ = πρ2 (9.5)

Vol. 49 (1998) Stieltjes function expanded at zero and infinity 151

λe(x)/λ1 − 1 = [π(d− 1)− 1]− 2πd(d− 1) ln(d)1x

+O1x

2,

d =√π/(π − 4Φ).

(9.6)

On account of Th. 8.1 and (4.3) − (4.4), for p = 2 and k = 2 the second orderbounds on λe(x)/λ1 are given by:

1 +xG1

1 + xG2 +xG3

1 + xG4≥ λeλ1≥ 1 +

xG11 + xG2 +

xG31 + xG4+

xV51

, −1 < x ≤ 0,

(9.7)

1 +xG1

1 + xG2 +xG3

1 + xG4≤ λeλ1≤ 1 +

xG11 + xG2 +

xG31 + xG4+

xV51, x ≥ 0. (9.8)

The coefficients Gn (n = 1, 2, 3, 4) and V5 appearing in (9.7)− (9.8) evaluated forthe volume fractions Φ = 0.78, 0.785, 0.7853, by recurrence formula (A.1) and(A.3), are depicted in Table 1.

Table 1. The coefficients Gn (n = 1, 2, 3, 4) and V5 of the continued fraction representation ofPade approximants [2/2]2 and [2/2]2 to the effective conductivity λe(x)/λ1.

Φ G1 G2 G3 G4 V50.7800 0.780000 0.023109 0.086890 0.202388 0.3562970.7850 0.785000 0.005798 0.101705 0.105038 0.4088190.7853 0.785300 0.002836 0.104514 0.081119 0.425574

The estimations [2/2]2 (left hand side of (9.7)−(9.8)) and [2/2]2 right hand sideof ((9.7) − (9.8)) of λe(x)/λ1 computed by formula (A.4) as well as the bounds[2/2]1 and [2/2]2 reported in [21,22] are depicted in Fig.4. For comparison theexact values of λe(x)/λ1 obtained in [25] are additionally drawn. It is worthnoting that for Φ = 0.78, Φ = 0.785, 0.7853 the lower bounds on λe(x)/λ1 agreewith high accuracy with the exact values of λe(x)/λ1. The estimations of λe(x)/λ1evaluated in this paper (Th. 8.1) are more narrow then those previously obtainedin [24, 25], cf. Fig.4.

10. Summary and conclusions

Special continued fractions (4.3)− (4.4), constructed for Stieltjes series developedat zero and infinity, have been exploited to prove that for fixed k the necessaryand sufficient conditions for the convergence of one-point Pade approximants to aStieltjes function [1,Th.16.2] apply also to two- and three-point Pade approximantsdenoted by [M/M ]k and [M/M ]k respectively. In a real domain the approximants[M/M ]k and [M/M ]k form monotone sequences of bounds converging to a Stieltjesfunction. Those approximants yield the best upper and lower bounds on Stieltjesfunctions provided that the radius of convergence R, p coefficients of the series(2.2), k terms of the series (2.4) and the value of U cf.(2.1) are given.

152 S. Tokarzewski and J. J. Telega ZAMP

General recurrence formulae (A.1)-(A.4) for finding the upper and lower es-timations of a Stieltjes function from the power series (2.2) and (2.4) have beenderived in the Appendix and tested successfully for correctness, cf.Figs 1,2.

The main result of this paper, formulated as Th. 8.1, establishes in terms of[M/M ]k and [M/M ]k the new bounds on the real effective transport coefficientsλe(x)/λ1 of two-phase media.

As an example of practical evaluations, the effective conductivity for a squarearray of equally-sized, highly conducting cylinders has been investigated in termsof second order bounds [2/2]2 and [2/2]2, see Fig.4.

Appendix

In this Appendix we propose the recurrence formulae for determination of theparameters:(i) Gn, n = 1, 2, ...2k

j = 1, 2, 3, ..., k , m = 2j − 1,

Gm = c(m)

1 , Gm+1 =c

(m)1

c(m)

0

, c(m+2)

0 = 1 , c(m+2)−1 = Gm+1,

c(m+2)n = − 1

c(m)

1

(n−1∑i=0

c(m+2)i c

(m)n+1−i

), n = 1, 2, ..., p− j,

(A.1)

c(m+2)n = − 1

c(m)

0

n−1∑i=−1

c(m+2)i c

(m)n−i

, n = 0, 1, ..., k − 1− j,

c(m+2)

1 := c(m+2)

1 −Gm+1 , c(m+2)

0 := c(m+2)

0 − 1,

(ii) gn, n = 2k + 1, ..., p+ k

m = 1, 2, ..., p− k , g2k+m = c′(m)1 ,

n = 1, 2, ..., p− k −m,

c′(m)0 = 1 , c

′(1+m)n = − 1

c′(m)1

(n−1∑i=0

c′(1+m)j c

′ (m)n+1−j

),

(A.2)

Vol. 49 (1998) Stieltjes function expanded at zero and infinity 153

(iii) Vp+k+1

V3 =U(1−G2R) +G1R

U,

V3 = 1−RG2 , Vj+2 =Vj(1−Gj+1R)−GjR

Vj, j = 3, 5, ..., 2k − 1,

(A.3)

V2k+1+j =V2k+j − g2k+jR

Vj, j = 1, 2, ..., p− k.

From the coefficients cn (n+1, 2, . . . , p) of the power expansion (2.2) and Cn (n =0, 1, . . . , k − 1) of the power series (2.4) the formulae (A.1)-(A.3) provide theparameters Gn (n = 1, 2, . . . 2k), gn (n = 2k + 1, . . . , p + k) and Vp+k+1for thecontinued fractions [M/M ]k and [M/M ]k,cf.(4.3)− (4.4).The recurrence formula for approximants [M/M ]k and [M/M ]k

Vp+k+1 = 0 for [M/M ]k , Q(0) = xVp+k+1\R,

Q(j+1) =xgp+k−j

1 +Q(j), j = 0, 1, ...p− k − 1,

Q(j+1) =xG2p−2j−1

1 + xG2p−2j +Q(j) , j = p− k, ..., p− 1 , Q(p) = [M/M ]k

(A.4)

complete the set of algorithms for computing the Pade approximants bounds on aStieltjes function xf1(x).

Acknowledgment

This work was supported by the State Committee for Scientific Research (Poland)through the Grants No 3 P404 013 06 and No 7 T07A 016 12.

References

[1] G. A. Baker, Essentials of Pade Approximants, Acad. Press, London 1975.[2] G. A. Baker and P. Graves-Morris, Pade Approximants. Part I: Basic Theory, in: Gian-

Carlo Rota, Ed., Encyclopedia of Mathematics and its Applications, 13, Addison-WesleyPubl. Co., London 1981.

154 S. Tokarzewski and J. J. Telega ZAMP

[3] G. A. Baker and P. Graves-Morris, Pade Approximants, Part II: Extensions and Appli-cations, in: Gian-Carlo Rota, Ed., Encyclopedia of Mathematics and its Applications, 14Addison- Wesley Publ. Co., London 1981.

[4] D. J. Bergman, The dielectric constant of a composite material – A problem in classicalphysics, Phys. Report. C43 (1978), 377–407.

[5] D. J.Bergman, Bounds for the complex dielectric constants of two-component compositematerial, Phys. Rev. B23(6) (1981) 3058–3065.

[6] D. J. Bergman, Rigorous bounds for the complex dielectric constants of a two-componentcomposite material, Ann.Phys. 138 (1982), 78–114.

[7] D. J. Bergman, Hierarchies of Stieltjes functions and their application to the calculation ofbounds for the dieletric constant of a two-components composite medium, SIAM J. Appl.Math. 53 (1993), 915–930.

[8] O. P. Bruno, The effective conductivity of an infinitely interchangeable mixture, Comm.Pure Appl. Math. XLIII (1990), 769–807.

[9] K. E. Clark and G.W.Milton, Optimal bounds correlating electric, magnetic and thermalproperties of two-phase, two-dimensional composites, Proc. Roy. Soc. London A 448(1995), 61–190.

[10] A. Bultheel, Laurent Series and their Pade Approximants, Birkhauser-Verlag, Basel 1987.[11] L. Casasus and P.Gonzalez-Vera, Two-point Pade type approximants for Stieltjes func-

tions, in: C. Brezinski, A. Draux, A.P. Magnus, P. Moroni and A. Ronveaux, Eds, Proc.Polinomes Ortogonaux et Applications, Bar-le-Duc, 1984, Lecture Notes in Math. 1171408–418, Springer, Berlin 1985.

[12] J. Gilewicz, Approximants de Pade Springer-Verlag, Berlin 1978.[13] K. Golden and G. Papanicolaou, Bounds for effective parameters of heterogeneous media

by analytic continuation, Comm.Math.Phys. 90 (1983), 473–491.[14] P. Gonzalez-Vera, O. Njastad, Convergence of two-point Pade approximants to series of

Stieltjes, J. Comp. Appl. Math. 32 (1990), 97–105.[15] W. B. Gragg, Truncation error bounds for T- fractions, in: E. W. Cheney, Ed., Approxi-

mation Theory III, 455–460, Academic Press, New York 1980.[16] Z. Hashin, S. Shtrikman, A variational approach to the theory of the effective magnetic

permeability of multiphase materials, J. Appl. Phys. Sol. 33 (1962), 3125–3131.[17] W. B. Jones, O. Njastad and W.J.Thron, Two-point Pade expansions for a family of analytic

functions, J. Comp. Appl. Math. 9 (1983), 105–123.[18] W. B. Jones and W. J. Thron, A posterior bound for truncation error of continued fraction,

SIAM J. Num. Anal. 8 (1970), 693–705.[19] W. B. Jones and W. J. Thron, Continued Fraction Analytic Theory and its Applications,

in: Gian-Carlo Rota, Ed., Encyclopedia of Mathematics and its Applications, 11, Addison-Wesley Publ. Co., London 1980.

[20] R. C. McPhedran and G. W. Milton, Bounds and exact theories for the transport propertiesof inhomogeneous media, Appl. Phys.A 26 (1981), 207–220.

[21] R. C. McPhedran, L. Poladian and G. W. Milton, Asymptotic studies of closely spacedhighly conducting cylinders, Proc.R.Soc.London A 415 (1988), 185–196.

[22] G. W. Milton, Bounds on the complex permittivity of a two- component composite material,J. Appl. Phys. 52 (1981), 5286–5293.

[23] G. W. Milton, Bounds on the transport and optical properties of two-component material,J. Appl. Phys. 52 (1981), 5294–5304.

[24] S. Tokarzewski, J. B lawzdziewicz and I. V. Andrianov, Two- point Pade approximants forformal Stieltjes series, Num. Algorithms 8 (1994), 313–328.

[25] S. Tokarzewski, J. B lawzdziewicz and I.V.Andrianov, Effective conductivity of denselypacked highly conducting cylinders, Appl. Phys.A 59 (1994), 601–604.

[26] S. Tokarzewski and J. J. Telega, Inequalities for two-point pade approximants to the ex-pansions of Stieltjes functions in real domain, Acta Appl. Math. (1997), in press.

Vol. 49 (1998) Stieltjes function expanded at zero and infinity 155

[27] H. S. Wall, Analytic Theory of Continued Fractions, New York,Van Nostrand 1948.[28] O. Wiener, Abhandlungen der mathematisch-physischen Klasse der koniglichen sachsischen

Gesellschaft der Wissenschaften, Abh. Sachs. Akad. Wiss. Leipzig Math.- Naturwiss. Kl.,32: 509, 1912.

S. TokarzewskiPolish Academy of SciencesInstitute of FundamentalTechnological ResearchSwietokrzyska 21, 00-049 WarsawPoland(Fax: +48 2699815)

J. J. TelegaPolish Academy of SciencesInstitute of FundamentalTechnological ResearchSwietokrzyska 21, 00-049 WarsawPoland(Fax: +48 2699815)

(Received: May 20, 1996; revised: April 28, 1997)