ABELIAN THEOREMS FOR THE STIELTJES TRANSFORM OF FUNCTIONS

An initial (final) value Abelian theorem concerning transforms of functions is a result in which known behavior of the function as its domain variable approaches zero (approaches =) is used to infer the behavior of the transform as its domain variable approaches zero (approaches ). We obtain such theorems in this paper concern%ng the StieltJes transform. In our results all parameters are complex; the variable s of the transform is complex in the right half plane; and the initial (final) value Abelian theorems are obtained as sl / 0 (Isl / =) within an arbitrary wedge in the right half plane.

In this section we obtain results which we shall use in the proofs of our Abelian theorems.The following result extends [i, Lemma] and [4, p. 40, Lemma 1.5].The proof which we give here is based on properties of Carlson's R function [6, p. 97, Definition 5.9-1; p. 137, Corollary 6.3-4; and p. 153, Theorem 6.8-1].
We correct an error and misprints in the proof of [i, Lemma].In [i, Lemma] we assumed that 0 and N are real numbers such that -i < N < 0. Thus because of the possible values for O and N a more complicated contour should have been chosen with which to apply the Cauchy theorem in the proof.Under the assumption that m Im(s) > 0, s CO, in the proof of [I, Lemma], we obtain the first equality iO' of [i, (3)] as before.Now let a re and A Re@', @' Arg(i/s), 0 < r < R.
Let F I denote the straight line segment from a to A; let F 2 be the arc of circum- ference z Re io, 0' _< 0 _< O, from z A to z R; let F 3 be the straight line segment along the real axis from z R to z r; and let F 4 be the arc of circum- ference z re 8'< 8 < 0, from z r to z a. Finally let F be the union of FI, F2, F3, and F 4. Cauchy's theorem can now be applied with respect to F and the Integrand (zn(l+z)-P-l), and we obtain x (l+x)-0-1 dx I + I + f z (I+z)-P-I dz z x+iy" F I F 2 F 4 Straightforward estimates show that lim I z (l+z) -0-I dz 0 lim I z N (l+z) -0-I dz r+0+ F R F 4 2 (2.4) Hence upon letting r / O+ and R / in (2.4), the proof is completed for the case --Im(s) > 0, s e C 0 as in [i, Lemma] using the first equality of [i, (3)].The proof for the case Im(s) < 0 in [i, Lemma] proceeds analogously.
The following two lemmas contain some inequalities which we shall use later.

P Pl + iP2
i + i 2 and s + i are complex numbers in these lemmas and throughout the remainder of the paper.
LEMMA 2.2.Let t > 0 be a real number.Let p, , and s be complex numbers such that p e C> + I e C> and s e C> We have n I (2.5) l(s+t)-P-ll < Is+tl -pI-I exp(Ip21/2 <_ (+t) -pI-I exp(nlp21/2) Further, if t > y > 0 for fixed y > 0 then ls+l -p-< =-P-(2.9) PROOF.All of the proofs follow easily by using the properties of the prln- clpal value power.As an example we prove (2.8) and leave the proofs for (2.5) (2.7) and (2.9) to the interested reader.We have sP-nl for s e C> and (2.8) is obtained.

I.
Let s e C> and let n be any complex number.We have II.Let n be a fixed complex number such that -I < N I Re(n) < 0 and let s e C> such that o Re(s) > i and n(o) > In ctn(n)l.We have III.Let n be a fixed complex number such that -I < n I Re(n) < 0 and let that 0 < < and l nl l >
r > 0 we know that ln(r) / as r -* O+ or as r / .Thus For real the assumptions n()> Jn ctn(n)l in II, lnlsll > In ctn(n)[ in III, and ln(o(l + K2)I/2) > In ctn(Nn) in IV are meaningful assumptions on Re(s) and s for fixed N. Now the proof of (2.10) is like those used to prove Lemma 2.2 and is left to the reader.To prove (2.11) we note that under the stated hypotheses in > enlsl I ctn(n)l > %n(o) l ctn(N)l (2.14) and all differences are positive.(2.11) follows by taking reciprocals in (2.14).
We shall need the Stieltjes transform formula contained in [7, p. 218, (28) in section 4 of this paper, and we state this formula in the following lemma.
LEMMA 2.4.Let N be a complex number such that -i < Re(N) < 0. Let s be a complex number such that IArg(s) < .Then t n(t) (s+t) -I dt s -ctn()) csc (T) (Log(s) The next lemma contains representations of two improper Riemann integrals which we shall need in our analysis in section 4. LEMMA 2.5.For 0 < < i and -I < 8 < 0 we have PROOF.First note that all series on the right of (2.17) and (2.18) are convergent.We prove (2.17) now.The improper integral n(t) (O+t) dt 0 0 < o < i, is defined to be the value of |O t8 n(t) (O+t) -I dt llm Furthermore, since the integral of a Riemann integrable function is a continuous function of its upper limit of integration [8, Theorem 7.32, pp. 161-162], we have Hence consider the integral over the interval 6 < t < oin (2.19).For such t we note that t/ol <_ (0-)/o < i; hence the series k--O converges uniformly for < t < oby the Welerstrass M-test [8, Theorem 9.6, p. 223].The interchange of integration and summation in the second llne of the following computation is therefore justified [8, Theorem 9.9, p. 226].
t 8 n(t) dt 0 -1 o-t 8 n(t) The last step above is justified since all four series are convergent by the alternating series test [8, Theorem 8.16, p. 188].By Abel's theorem [8, Theorem 9.31, p. 245] we know that we can evaluate the limit as / 0+ in (2.20) by merely setting 0 in the last equality.Also since the last two series in (2.20) converge uniformly in 8, say for 0 < < o/2, and since 8 + 1 > 0, then the product involving the last two series in (2.20) converges to zero as 6 / 0+.
The proof of (2.18) is completely analogous to that of (2.17).We split the integral in (2.18) at o, i < o < m, and proceed similarly as in the proof of (2.17).We leave the now straightforward details to the interested reader.The proof of Lemma 2.5 is complete.
In the Abelian theorems for functions of [i] [4], hypotheses are placed on the quotient f(t)/t for certain specified real and then limit properties are obtained for the generalized Stieltjes transform of f(t).In this section we allow to be complex in the assumptions on the quotient f(t)/t; the Abelian theorems for functions of [i] [4] become special cases of the results presented in this section.We note that there exist complex valued functions f(t) of the real variable t > 0 which satisfy the hypotheses stated in both Theorem 3.1 and Theorem 3.2 below.
Let K > 0 be an arbitrary but fixed real number.We recall the wedge PK {s s q + i, o > O, II < Ko} in C> as defined in Lemma 2.3 (IV).Our initial value Abelian theorem is as follows.
THEOREM 3.1.Let p and be complex numbers such that p e C> and D + i e C> Let f(t) be a complex valued function of the real variable t > 0 such that the generalized StleltJes transform F(s) of f(t) exists for s e C> and such that (f(t)/t) is bounded on y < t < for all y > 0. Let e be a complex number such that lim f(t) PROOF.Using (i.I) and Lemma 2.1 we have for any y > 0 that ( We first estimate I I. Let > 0 be arbitrary.Applying the hypothesis (3.1), there exists a 6 > 0 such that f(t) t < E if 0 < t < We now fix y in (3.3) such that 0 < y < 6 and obtain and the supremum is finite since by hypothesis (f(t)/tr]) is bounded on y < t < m.Now use (2.5), the first inequality of (2.6), and (2.8)  Recall that p -C> so that Pl ing the integration we have I > 0. Using (2.9) in (3.9) and then perform- (3.i0) is valid for all s g C> and shows that 12 / 0 as sl / 0 in any manner in C>.We now combine (3.3), (3.7), and (3.10) to obtain lira sup sl+0 Is p-F(s) B(p-q,r+l) < exp(lp2i/2) exp(Ip2-r]21/2 B(Pl-nl,r]l+l) (i + K2) (pl-r]l)/2 sgP K where > 0 is arbitrary.The desired result (3.2) follows immediately, and the proof is complete.
The approach of s to zero inside the wedge PK for arbitrary but fixed K > 0 in Theorem 3.1 is a sufficient condition for the desired conclusion (3.2) to hold but is not a necessary condition.The example of [I, p. 51] shows this.
We now prove our final value Abelian theorem.THEOREM 3.2.Let p and q be complex numbers such that p q E C> and C> Let f(t) be a complex valued function of the real variable t > 0 such that the generalized Stieltjes transform F(s) of f(t) exists for s e C> and such that (f(t)/t) is bounded on 0 < t < y for all y > 0. PROOF.We begin with (3.3) exactly as in the proof of Theorem 3.1 where I I and 12 are exactly as in (3.3).For arbitrary > 0 we apply the hypothesis (3.11) and choose a fixed y > 0 large enough to obtain (3.13)Using (2.5), the second inequality of (2.6), and (2.8) in (3.13) we get 12 < exp(IP21/2) exp(IP2-n21/2) Isl PI-I y tl Putting (3.15) into (3..14) and restricting s e C> to PK K > 0 being arbitrary but fixed, we have PI-I 12 < exp(Ip21/2) exp(Ip2-21/2) (i + K) B (pl-r I rl+l) (3.16) and > 0 is arbitrary here.
Using the boundedness hypothesis of (f(t)/tN), (2.5), the second inequality of (2.7), and (2.8)  + I e C> A combination of (3.17) and (3.18)  limit superior argument as in the proof of Theorem 3.1.The proof is complete.
The final value Abelian theorems [i, Theorem 2], [2, pp. 183185], [3, Theorem 4.1.i],and [4, Theorem 4.2] are special cases of Theorem 3.2.Notice that the conclusion of [i, Theorem 2] was obtained for Isl / oo, s e S K {s s G + i, > 0, lwl < K}, where K> 0 is arbitrary but fixed; that is s was allowed to get big in absolute value within a strip centered about the real axis in C> The conclusion of our present Theorem 3.2 allows Is] to get big within a wedge PK in C> which is a more general situation than [i, Theorem 2].
For the special case of Theorem 3.2 considered [4, Theorem 4.2], Tiwari also recognized that S K could be replaced by PK in his result.

FURTHER ABELIAN THEOREMS.
In a private communication to one of us, R.D.C., Lavoine [9] suggested that an attempt be made to replace the assumption of the type (f(t)/t) / a as t + 0+ or as t / in Abelian theorems for the Stieltjes transform by the more general where j is a positive integer and n(t) is the natural logarithm.In a recent paper Lavoine and Misra [i0] have obtained Abelian theorems for the distributional Stieltjes transform in which an assumption of the type (4.1) is made on the distri- butions under consideration.The Stieltjes transform of the distributions is con- sidered for the case that the variable s of the transform is real, and the Abelian theorems are then obtained as s / 0+ or s / .In this section we shall obtain Abelian theorems for functions under an assumption like (4.1) in which the variable s of the Stieltjes transform is in C> and the results are obtained as sl + 0+ or sl / in a wedge PK We note that there exist functions f(t) which satisfy the hypotheses of Theorems 4.1 and 4.2.
Our initial value result is as follows.
THEOREM 4.1.Let N be a complex number such that -i < ql Re(H) < O. Let f(t) be a complex valued function of the real variable t > 0 such that the Stieltjes transform F(s) 0 f(t) (s+t) -I dt exists for s g C> and such that (f(t)/t q n(t)) is bounded on y < t < for all y > 0. Let e be a complex number such that lim f(t) t+0+ t q n(t) ( We first estimate I I. Let > 0 be arbitrary; by (4.2) there exists a such that (4.5) Now fix y > 0 such that 0 < y < min{l,}.In this result we are letting sl + 0, s e PK for K > 0 arbitrary but fixed Hence to obtain our result it suffices to assume that 0 < sl < i and lnlsll > IT ctn(DT) and that for K > 0 fixed, ((i + K2) I/2) < i and lln(o(l + K2)I/2)] > 17 ctn(N)l, where a Re(s) > 0. We emphasize that we are making the assumptions on s e PK as stated in the preceding sentence throughout the remainder of this proof.Thus for s g PK and the above fixed y > 0 we apply (4.6), (2.5), the second inequality of (2.6) with p 0, (2.10), and (2.13) to obtain exp(In21/2) f'l tnl ln(t)l dr.Isl (ln(o(I+K2) )I -17 ctn (7) l) Now 0 < Is < i implies 0 < q Re(s) < i; and we know that ln(t) -In(t), 0 < t < I. Thus we use (2.17 where -i < NI Re(N) < 0 here.Using (4.9) we have is i (ln(o(l + K2)I/2)I i ctn(n)l) l)k k=O n 1-k (4.11)Further, it is easy to see that the absolute value of J2 J3 and J5 when divided by the denominator on the left of (4.10) and (4.11) all tend to zero in the limit as Isl 0, s c PK" Using this fact together with (4.8), (4.10), and Let us now estimate 12 in (4.5).Take y > 0 to be fixed as in the sentence succeeding (4.6) so that 0 < y < min{l,6}.Applying the boundedness hypothesis for (f(t)/t n(t)) on y < t < , (2.5)  Recall that we are assuming the conditions on s e PK under which (2.13) holds without loss of generality in this proof.follows.We now combine (4.4), (4.12), and (4.16) to conclude that for any fixed Since > 0 is arbitrary on the right of (4.17), the conclusion (4.3) follows.
The proof is complete.
We now obtain a similar final value Abelian theorem.
THEOREM 4.2.Let N be a complex number such that -i < Nl Re(H) < O. Let f(t) be a complex valued function of the real variable t >_ 0 such that the Stieltj es transform F(s) 0 f(t) (s+t) -I dt exists for s e C> and such that (f(t)/t N n(t)) is bounded on 0 < t < y for all y > 0. Let e be a complex number such that lim f(t) .PROOF.Let K > 0 be arbitrary but fixed.In this result we are letting Isl / s e PK to obtain (4.19).As Isl / oo, s g PK @ Re(s) must tend to also.Thus without loss of generality we assume throughout this proof that o Re(s) > i and n(o) > In ctn(nn) l.Now proceeding as in the proof of Theorem 4.1 we obtain (4.4) where I I and 12 are defined in (4.5).To estimate 12 we first take an arbitrary > 0 and apply hypothesis (4.18)  (4.20) For y > i so fixed we apply (4.20), (2.5), the second inequality of (2. [KI;,+ K 2 + K 3 + K 4 + K 5] and [K I + K 2 + K 3 + K 4 + K 5] is positive.For s E PK we use (4.9) and L'Hospital's Using (4.9) it is easy to see that the absolute value of K 2 and K 4 when divided by the denominator on the left of (4.23) and (4.24) both tend to zero as Isl -,  since we assumed at the beginning of this proof that o > i. Recalling that -i < n I < 0, (4.26) proves that the left side of (4.26) tends to zero as Is / , s e PK We thus conclude from (4.22), (4.23), (4.24), (4.26), and the facts stated above concerning K 2 and K 4 We now estimate I I.For the y > I fixed in (4.20)Using the boundedness hypothesis for (f(t)/t n(t)), (2.5), a proof as in obtain- ing (2.10), (2.11), and taking the limit in (4.28) as 8 / O+ we obtain
y < min{l,} and -i < NI < 0 ctn(n)I) o(gm() I'n" ctn(n)I) " =tn(r)I) k=O (l+k+r I) Let be a complex to obtain a fixed y > I such that f(t)In(t)< if t > y > i.
and any such that