PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 67(81) (2000), 31–43

REGULARLY VARYING SEQUENCES AND ENTIRE FUNCTIONS OF FINITE ORDER

Slavko Simić

Communicated by Stevan Pilipović

Abstract. We present a method for estimating the asymptotic behavior of:

f^{α}(x) := ∑_{n=1}^{∞} n^{α}l_{n}a_{n}x^{n}, x → ∞, α ∈ R,

related to a given entire function f(x) := ∑_{n=1}^{∞} a_{n}x^{n} of finite order ρ, 0 < ρ < +∞, a_{n} ≥ 0, n ∈ N; where (l_{n}), n ∈ N, are slowly varying sequences in Karamata’s sense.

Preliminaries

A. Slowly varying functions l(x) in Karamata’s sense are defined on a positive part of real axis, positive, locally bounded and satisfy: lim_{x→∞} \frac{l(λx)}{l(x)} = 1, for each λ > 0.

The class R_{α} of regularly varying functions (r.v.f.) with index α consists of all functions a(x) which can be represented as: a(x) = x^{α}l(x), for some α ∈ R.

The theory of r.v.f. is very well developed and an excellent survey of results is given in [1] and [3].

Here we put special attention on a class SR_{α} ⊂ R_{α} (smoothly varying functions; [1, p. 44]) i.e., b(x) ∈ SR_{α} if it is a C^{∞} r.v.f. of index α, satisfying

x^{n}b^{(n)}(x)/b(x) → α(α − 1) ⋯ (α − n + 1) = (α)_{n}, x → ∞, n ∈ N.

Some important properties of this class are:

If f ∈ SR_{α}, g ∈ SR_{β}, then

f · g ∈ SR_{α+β}; f ∘ g ∈ SR_{αβ}; f′ ∈ SR_{α−1}, α ∈ R^{+}.

Also, for a given c(x) = x^{−α}l(x), α ∈ R^{+}, we consider its dual c^{∗}(x) defined by

c^{∗}(x) := \frac{1}{Γ(α)} ∫_{0}^{∞} e^{−xy} \frac{c(1/y)}{y}dy = \frac{1}{Γ(α)} ∫_{0}^{∞} e^{−xy}y^{α−1}l(1/y) dy; x, α ∈ R^{+}.

The next proposition is of crucial importance.

1991 Mathematics Subject Classification. Primary 30D20, 26A12.

Proposition 1. We have: c^{∗}(x) ∈ SR_{−α}; c^{∗}(x) ∼ c(x), x → ∞.

Proof. That c^{∗}(x) ∼ c(x), x → ∞, is a consequence of Karamata’s Tauberian theorem for Laplace transforms (with 0^{+} and ∞ reversed) (cf. [1, p. 43]).

By the same argument

(c^{∗}(x))^{(n)} = \frac{(−1)^{n}}{Γ(α)} ∫_{0}^{∞}e^{−xy}y^{α+n−1}l(\frac{1}{y} ) dy ∼ \frac{(−1)^{n}Γ(α + n)}{Γ(α)}x^{−(α+n)} l(x), x→∞.

Hence,

\frac{x^{n}(c^{∗}(x))^{(n)}}{c^{∗}(x)} ∼ \frac{(−1)^{n}Γ(α + n)}{Γ(α)} = (−α)_{n}, x → ∞;

i.e., c^{∗}(x) ∈ SR_{−α}.

We could treat regularly varying sequences (r.v.s.) as r.v.f. defined on N (see [2]) i.e., (a_{n}) is a r.v.s. with index α if it has the form

a_{n} = n^{α}l_{n}; l_{n} = l(n), n ∈ N, α ∈ R,

for some slowly varying function l(x) defined for x ∈ R^{+}.

Examples of l_{n} are:

ln^{a} 2n, ln^{b}(ln 3n), e^{(ln n)^{c}}, e^{\frac{ln 2n}{ln(ln 3n)}} ,…; a, b ∈ R, 0 < c < 1.

B. Denote by G := {g | g : R^{+} → R^{+}, g ∈ C^{1}}, and define there an operator ĝ,

ĝ(x) := \frac{xg′(x)}{g(x)}.

Some properties of this operator are (g, h ∈ G):

\begin{matrix}1. ĉg = ĝ, c ∈ R^{+}; & 5. { \atop g^{a} · h^{b}} = aĝ + bĥ, a, b ∈ R; \\ 2. x̂^{a} = a, a ∈ R; & 6. { \atop g ∘ h} = (ĝ ∘ h) · ĥ; \\ 3. { \atop g + h} ≤ max(|ĝ|, |ĥ|); & 7. ĝ ∈ G ⇒ g ↑, x ∈ R^{+}; \\ 4. ĝ · h = ĝ + ĥ; & 8. (ĝ(x) → α, x → ∞, α ∈ R) ⇒ g ∈ R_{α}. \end{matrix} (1)

We also consider a set of entire functions F (z) = ∑_{k=0}^{∞} a_{k}z^{k}, with non-negative coefficients and of finite order ρ, 0 < ρ < ∞. By definition:

ρ = lim sup_{x→∞} \frac{ln ln M_{F} (x)}{ln x}

where M_{F} (x) denotes the maximum modulus of F (z) on the circle |z| = x. In our case we have:

M_{F} (x) = max_{|z|=x} |F (z)| = max_{|z|=x} | ∑_{k=0}^{∞} a_{k}z^{k}| = ∑_{k=0}^{∞} a_{k}x^{k} = F (x), x ∈ R^{+}.

Let us deenote: f(x) := M_{F} (x) = (F (x), x ∈ R^{+}). Hence:

lim sup_{x→∞} \frac{ln ln f(x)}{ln x} = ρ, ρ ∈ R^{+},  (2)
f ∈ C^{∞}; f^{(n)} ∈ G; f̂ ∈ G.

REGULARLY VARYING SEQUENCES AND ENTIRE FUNCTIONS 33

Proposition 2. We have ˆ f̂ ∈ G.

Proof. Taking into account properties (1) and (2), we have:

ˆf̂ = x̂f′ − f̂ = \frac{1}{f̂}(\frac{x(xf′)′}{f} − f̂^{2}).

Since

x(xf′)′ = ∑_{k=0}^{∞} k^{2}a_{k}x^{k}; f̂ = \frac{1}{f} ∑_{k=0}^{∞} ka_{k}x^{k},

we obtain:

ˆf̂ = \frac{1}{ff̂} ∑_{k=0}^{∞}(k − f̂)^{2}a_{k}x^{k} > 0, x ∈ R^{+};

and the proof is over.

Corollary 1. The function f̂ is monotone increasing on R^{+}.

Proposition 3. We have: lim sup_{x→∞} \frac{ln f̂(x)}{ln x} = ρ.

Proof. Let

δ := lim sup_{x→∞} \frac{ln f̂(x)}{ln x}, δ ∈ R^{+}.

From (2) it follows that, for each positive ϵ and large enough x:

ln f(x) < x^{ρ+ϵ}, x > x_{0}.

Corollary 1 gives

ln f(ex) − ln f(x) = ∫_{x}^{ex} \frac{f′(t)}{f(t)} dt = ∫_{x}^{ex} f̂(t) · \frac{dt}{t} > f̂(x) ∫_{x}^{ex} \frac{dt}{t} = f̂(x),

hence, for x > x_{0}, we get:

f̂(x) < ln f(ex) < (ex)^{ρ+ϵ},

i.e.,

\frac{ln f̂(x)}{ln x} < (ρ + ϵ)(1 + 1/ ln x), x > x_{0}.

Since ϵ is arbitrarily small, we conclude δ ≤ ρ.

From the other side ln f̂(x) < (δ + ϵ) ln x for x > x_{1}, i.e.,

f̂(x) < x^{δ+ϵ}, \frac{f′(x)}{f(x)} < x^{δ−1+ϵ}; x > x_{1}.

It follows that

ln f(x) = ∫_{x_{1}}^{x} \frac{f′(t)}{f(t)} dt + ln f(x_{1}) < \frac{x^{δ+ϵ}}{δ + ϵ} + O(1), x > x_{1};

i.e.,

\frac{ln ln f(x)}{ln x} < δ + ϵ + o(1), x → ∞;

i.e.,

ρ ≤ δ.

Thus, we conclude:

lim sup_{x→∞} \frac{ln f̂(x)}{ln x} = lim sup_{x→∞} \frac{ln ln f(x)}{ln x} = ρ.

Corollary 2. The function f̂(x) is strictly increasing on R^{+} and lim_{x→∞} f̂(x) = +∞.

We also consider the set of entire functions {f_{m}} generated from f by the recurrence relation

f_{m}(x) := xf′_{m−1}(x), f_{0}(x) = f(x), m ∈ N.

They are of the same order ρ and evidently satisfy:

Proposition 4. f_{m}(x) = ∑_{k} k^{m}a_{k}x^{k}; f̂_{m}(x) ↑ ∞; ˆ f̂_{m}(x) > 0

f_{m} = f_{m−1}f̂_{m−1} = f ∏_{1}^{m} f̂_{k−1}; f̂_{m} = f̂_{m−1} + ˆ f̂_{m−1} = f̂+ ∑_{1}^{m} ˆ f̂_{m−1}.

Main results

Now we come to our main subject, i.e., the investigation of the asymptotic behavior concerning functions f^{α}(x) := ∑_{k=0}^{∞} c_{k}a_{k}x^{k}, related to a given entire function f(x) := ∑_{k=0}^{∞} a_{k}x^{k} considered before and where (c_{k}), (c_{0} := 1) is any regularly varying sequence of index α.

It is not difficult to prove that {f^{α}(x)} are also entire functions of the same order ρ as f(x) (using, for example, the relation: ρ = lim sup_{n→∞} \frac{n ln n}{ln(1/|a_{n}|)}).

The main idea of our method is to replace sequences (c_{k}) with asymptotically equivalent (c_{k}^{∗}) achieving thus an integral representation for f^{α}(x) (see also [5]). Then, using analytic properties of c^{∗}(x) and f(x), we establish the required asymptotic behavior in an almost elementary way.

Theorem A. If ˆ f̂(x) is bounded from above, then:

\frac{f^{α}(x)}{f(x)} ∼ c_{[f̂(x)]}, x → ∞,

for any regularly varying sequence (c_{k}) of index α, α < 0.

As we already explained, we first prove the theorem for a subclass of r.v.s. generated by c^{∗}(x) ∈ SR_{α}, i.e.,

Proposition A1. Theorem A is valid for sequences (c_{k}^{∗}) defined by

c_{k−1}^{∗} := c^{∗}(k), k ∈ N; c^{∗}(x) := \frac{1}{Γ(α)} ∫_{0}^{∞} e^{−xt}t^{α−1}l(1/t) dt, x ∈ R^{+}.

Proof. With: u(α, t) := \frac{1}{Γ(α)}t^{α−1}l(1/t) we produce an integral representation for:

f_{∗}^{(α)}(x) := ∑_{k=0}^{∞} c_{k}^{∗}a_{k}x^{k} = ∑_{k=0}^{∞} a_{k} ∫_{0}^{∞} e^{−t}u(α, t)(xe^{−t})^{k}dt =

= ∫_{0}^{∞} e^{−t}u(α, t)(∑_{k=0}^{∞} a_{k}(xe^{−t})^{k})dt = ∫_{0}^{∞} e^{−t}u(α, t)f(xe^{−t}) dt.

The interchanging of the sum and the integral is justified since both converge for x ∈ R^{+}. Now:

\frac{f_{∗}^{(α)}(x)}{f(x)} = (∫_{0}^{ξ} + ∫_{ξ}^{∞})(e^{−t}u(α, t)\frac{f(xe^{−t})}{f(x)} dt) = T_{1} + T_{2}

where ξ = ξ(x) := f̂(x)^{−1/2}.

For estimating T_{1} we use the following identity:

ln \frac{f(xe^{−t})}{f(x)} + tf̂(x) = ∫_{0}^{t} wf̂(a) ˆf̂(a) dw, a := xe^{w−t}.

Taking into account Proposition 2 and condition from Theorem A, we have:

0 < ˆ f̂(a) ≤ M < +∞,

where the constant M does not depend on a.

Also, since a ≤ x, Corollary 1 gives f̂(a) ≤ f̂(x), i.e.,

0 < ∫_{0}^{t} wf̂(a) ˆf̂(a) dw ≤ Mf̂(x) ∫_{0}^{t} w dw = \frac{M}{2} f̂(x)t^{2}.

Hence,

ln \frac{f(xe^{−t})}{f(x)} = f̂(x)(−t + O(t^{2})), x ∈ R^{+}, t ≥ 0

where the constant in O is independent of x or t. From here it follows

T_{1} = ∫_{0}^{ξ} e^{−t}u(α, t) exp(ln \frac{f(xe^{−t})}{f(x)}) dt = ∫_{0}^{ξ} e^{−t}u(α, t)e^{−tf̂(x)}e^{O(f̂(x)t^{2})}dt.

Since, for any B ∈ R^{+}, e^{B} = 1 + O(Be^{B}) and, for t ∈ (0, ξ), f̂(x)t^{2} = O(1), we obtain:

T_{1} = ∫_{0}^{ξ} e^{−t}u(α, t)e^{−tf̂(x)} dt + ∫_{0}^{ξ} e^{−t}u(α, t)e^{−tf̂(x)}O(f̂(x)t^{2}) dt =
= ∫_{0}^{∞} e^{−t(1+f̂(x))}u(α, t) dt − ∫_{ξ}^{∞} e^{−t}u(α, t)e^{−tf̂(x)} dt
+ O(f̂(x)) ∫_{0}^{∞} t^{2}u(α, t)e^{−t(1+f̂(x))} dt = T_{11} + T_{12} + T_{13}.

Now:

T_{11} = c^{∗}(f̂(x) + 1) ∼ c_{[f̂(x)]}^{∗}, x → ∞;
|T_{12}| = O(e^{−ξf̂(x)} ∫_{0}^{∞} e^{−t}u(α, t) dt) = O(e^{−f̂(x)^{1/2}});
T_{13} = O(f̂(x)) · \frac{d^{2}c^{∗}(s)}{ds^{2}}_{[s=1+f̂(x)]}

= O(f̂(x)) · O(\frac{c^{∗}(s)}{s^{2}})_{[s=1+f̂(x)]} = O(\frac{c^{∗}(f̂(x))}{f̂(x)}),

since c^{∗}(s) ∈ SR_{α}. Hence, we conclude that: T_{1} ∼ c_{[f̂(x)]}^{∗}, x → ∞.

For the estimation of the integral T_{2} the next lemma is necessary.

Lemma A1. Under the condition of Theorem A, i.e., sup ˆ f̂(x) ≤ M < +∞, for each x, t ∈ R^{+}:

\frac{f(xe^{−t})}{f(x)} ≤ exp(\frac{e^{−Mt} − 1}{M}f̂(x)).

Proof. Write the condition as

D(ln f̂(s)) ≤ M D(ln s), s > 0.  (A1.1)

Integrating (A1.1) over [xe^{−u}, x], u ≥ 0, we obtain

f̂(xe^{−u}) ≥ f̂(x) · e^{−Mu}.  (A1.2)

Integrating (A1.2) for u ∈ [0, t], we come to the conclusion from the lemma. Therefore,

T_{2} = ∫_{ξ}^{∞} e^{−t}u(α, t)\frac{f(xe^{−t})}{f(x)} dt ≤ ∫_{ξ}^{∞} e^{−t}u(α, t) exp(\frac{e^{−Mt} − 1}{M}f̂(x)) dt

< exp(\frac{e^{−Mf̂(x)^{−1/2}} − 1}{M}f̂(x)) · ∫_{0}^{∞} e^{−t}u(α, t) dt;

i.e.,

T_{2} = O(e^{−f̂(x)^{1/2}}), x → ∞;

so, Proposition A1 is proved.

The assertion of Theorem A follows using the fact c_{n} ∼ c_{n}^{∗}, n → ∞ and a variant of Toeplitz’s Limit Preservation Theorem (cf. [8, p. 36]) which says:

Let {ϕ_{k}(x)}, k = 0, 1, 2,…, be a set of non-negative functions defined on R^{+}, satisfying ∑_{k} ϕ_{k}(x) = 1, and let (s_{k}), k = 0, 1, 2,… be any convergent sequence of positive reals, lim s_{k} = s.

Then a necessary and sufficient condition for ∑_{k} s_{k}ϕ_{k}(x) → s, x → ∞, is lim_{x→∞} ϕ_{k}(x) = 0, for each fixed k ∈ N.

We are going to use this proposition by putting:

ϕ_{k}(x) := \frac{c_{k}^{∗}a_{k}x^{k}}{f_{∗}^{(α)}(x)}; s_{k} := \frac{c_{k}}{c_{k}^{∗}} , k = 0, 1, 2,….

Then,

∑_{k} ϕ_{k}(x) = 1; s = 1; ∑_{k} s_{k}ϕ_{k}(x) = \frac{f^{(α)}(x)}{f_{∗}^{(α)}(x)},

and all we have to prove is lim_{x→∞} ϕ_{n}(x) = 0 for fixed n.

For a_{n} ≠ 0 (otherwise, there is nothing to prove) write

ϕ_{n}(x) = \frac{c_{n}^{∗}a_{n}x^{n}}{f_{∗}^{(α)}(x)} = c_{n}^{∗}(\frac{a_{n}x^{n}}{f(x/2)})(\frac{f(x/2)}{f(x)})(\frac{f(x)}{f_{∗}^{(α)}(x)}).

From Proposition A1:

\frac{f(x)}{f_{∗}^{(α)}(x)} ∼ 1/c_{[f̂(x)]}^{∗} = O(f̂(x)^{2|α|}).

Lemma A1, for t = ln 2, gives

\frac{f(x/2)}{f(x)} ≤ exp(−\frac{1 − 2^{−M}}{M} f̂(x)), M > 0;

and, evidently: f(x/2) > a_{n}(x/2)^{n}. Hence,

ϕ_{n}(x) = O(f̂(x)^{2|α|} exp(−\frac{1 − 2^{−M}}{M} f̂(x))) = o(1), x → ∞, f̂(x) ↑^{∞},

i.e.,

f^{(α)}(x) ∼ f_{∗}^{(α)}(x) ∼ f(x) c_{[f̂(x)]}^{∗} ∼ f(x) c_{[f̂(x)]}, x → ∞; α < 0;

therefore, Theorem A is valid.

Our task now is to extend the validity of Theorem A to non-negative indexes of r.v.s. (c_{k}). First of all, we prove

Proposition 5. Under conditions of Theorem A, for any α ≥ 0 we have

lim inf_{x→∞} \frac{f^{(α)}(x)}{(f̂(x))^{α}l(f̂(x))f(x)} ≥ 1.

Proof. We use a form of Hölder’s inequality:

∑_{k} u_{k}w_{k} ≥ (∑_{k} u_{k}^{p})^{1/p}(∑_{k} w_{k}^{q})^{1/q}, u_{k}, w_{k} ∈ R^{+}, \frac{1}{p} + \frac{1}{q} = 1, q < 0. (7.1)

Putting there

u_{k} = k^{−1}(l_{k}a_{k}x^{k})^{1/p}; w_{k} = \{ \begin{matrix}k^{α+1}(l_{k}a_{k}x^{k})^{1/q}, a_{k} > 0; \\ 0, a_{k} = 0 \end{matrix} , k ∈ N,

we obtain:

f^{(α)}(x) ≥ (f^{(−p)}(x))^{1/p}(f^{(q(α+1))}(x))^{1/q}.

Theorem A gives

lim inf_{x→∞} \frac{f^{(α)}(x)}{(f̂(x))^{α}l(f̂(x))f(x)} ≥

f ≥ lim_{x→∞}(\frac{f^{(−p)}(x)}{(f̂(x))^{−p}l(f̂(x))f(x)})^{1/p} · lim_{x→∞}(\frac{f^{(q(α+1))}(x)}{(f̂(x))^{q(α+1)}l(f̂(x))f(x)})^{1/q} = 1.

An extension of Theorem A is the following

Theorem A’. For any ϵ > 0, Theorem A is valid for α ≤ 2 − ϵ.

Proof. Put in (7.1):

u_{k}^{p} = k^{2−ϵ}l_{k}a_{k}x^{k}; w_{k}^{q} = \{ \begin{matrix}k^{2}a_{k}x^{k}, a_{k} > 0 \\ 0, a_{k} = 0 \end{matrix} , k ∈ N.

Since we could restrict 0 < ϵ ≤ 2, by taking p = ϵ/3, it follows that

∑_{k} k^{2−ϵ}l_{k}a_{k}x^{k} ≤ f_{2}^{−p/q}(∑_{k} k^{−1}l_{k}^{1/p}a_{k}x^{k})^{p}, k ∈ N.

But

f_{2} = ff̂f̂_{1} = ff̂(f̂+ ˆ f̂) ∼ ff̂^{2}, x → ∞,

and

∑_{k} k^{−1}l_{k}^{1/p}a_{k}x^{k} ∼ f̂^{−1}(l(f̂))^{1/p}f, x → ∞.

Hence,

lim sup_{x→∞} \frac{f^{(2−ϵ)}(x)}{(f̂(x))^{2−ϵ}l(f̂(x))f(x)} ≤ lim_{x→∞} \frac{(f(x)f̂(x)^{2})^{−p/q}(f̂(x)^{−1}(l(f̂(x)))^{1/p}f(x))^{p}}{(f̂(x))^{2−ϵ}l(f̂(x))f(x)} =1.

This together with Proposition 5 proves the theorem.

Therefore we see that the Theorems A and A’ provide the required asymptotic behavior of f^{(α)}(x) for all regularly varying sequences (c_{k}) with index less than 2.

Commentaries

The ondition sup ˆ f̂ < +∞ seems a little ambiguous but is not very restrictive, as we are going to show.

The explicit representation

f̂(x) = f(1) exp ( ∫_{1}^{x}\frac{ˆf̂(t)}{t} dt); 0 < ˆ f̂(t) ≤ M,

means that f̂ belongs to the class ER (Extended Variation, see [1, p. 74]).

Moreover, from ˆ f̂(x) = f̂_{1}(x) − f̂(x), we see that ˆ f̂ is of bounded variation (as a difference between two monotone increasing functions), i.e., bounded on finite intervals. Therefore, condition in Theorem A could be replaced by

lim sup_{x→∞} ˆ f̂(x) < +∞.

Strengthening this a bit, we obtain:

Proposition 6. If there exist lim_{x→∞} ˆ f̂(x) = δ, then δ = ρ and

lim_{x→∞} \frac{ln ln f(x)}{ln x} = lim_{x→∞} \frac{f̂(x)}{ln x} = lim_{x→∞} ˆ f̂(x) = ρ.

Moreover, in this case f̂ ∈ R_{ρ}.

Proof. This follows from Proposition 2 and

ˆf̂(x) = \frac{D(ln f̂(x))}{D(ln x)} = \frac{D(xD(ln f(x)))}{D(ln f(x))}.

The second part is Property 8 from (1) (cf. [1, p. 59]).

At this point we could connect asymptotically f̂ with ln f; namely:

Proposition 7. The following are equivalent:

(i) ln f(x) ∼ a x^{ρ}b(x); (ii) f̂(x) ∼ aρ x^{ρ}b(x), x → ∞; b(x) ∈ R_{0}, a, ρ ∈ R^{+}.

Proof. (ii) ⇒ (i): Since: ln f(x) = ∫_{1}^{x} f̂(t) · dt/t + O(1), the statement follows from r.v.f. Integration Theorem [3], i.e.,

ln f(x) ∼ aρ b(x) ∫_{1}^{x} t^{ρ} dt/t + O(1) ∼ a x^{ρ}b(x), x → ∞.

(i) ⇒ (ii): Corollary 2 gives, for x ≥ y > 0:

ln f(x) − ln f(y) = ∫_{y}^{x} f̂(t) dt/t\{ \begin{matrix}≤ f̂(x) ln x/y \\ ≥ f̂(y) ln x/y, \end{matrix}  (5.1)

Putting in (5.1) x = λy, λ > 1 and y = λx, λ < 1, we get

f̂(x)\{ \begin{matrix}≤ \frac{ln f(λx)−ln f(x)}{ln λ}, λ > 1 \\ ≥ \frac{ln f(x)−ln f(λx)}{ln 1/λ}, 0 < λ < 1. \end{matrix}

Therefore,

lim sup_{x→∞} \frac{f̂(x)}{a x^{ρ}b(x)} ≤ \frac{1}{ln λ}( lim_{x→∞} \frac{ln f(λx)}{a x^{ρ}b(x)} − lim_{x→∞} \frac{ln f(x)}{a x^{ρ}b(x)}) = \frac{λ^{ρ} − 1}{ln λ}, λ > 1; (5.2)

and analogously:

lim inf_{x→∞} \frac{f̂(x)}{a x^{ρ}b(x)} ≥ \frac{1 − λ^{ρ}}{ln 1/λ} , 0 < λ < 1.  (5.3)

Since the right-hand side does not depend on x, putting λ ↓ 1 in (5.2) and λ ↑ 1 in (5.3), we obtain the statement from Proposition 7.

Further extension needs some smoothnes condition on f, i.e.,

Theorem B. If ln f(x) ∈ SR_{ρ}, then

\frac{f^{(β)}(x)}{f(x)} ∼ ρ^{β} c_{[ln f(x)]}, x → ∞;

for any regularly varying sequence (c_{k}) of arbitrary index β ∈ R.

For justification of the condition from this theorem we cite an adapted version of Valiron’s Proximate Order Theorem i.e., (cf. [1, p. 311]):

If f is an entire function of finite order ρ, then there always exists a g ∈ SR_{ρ}, with:

lim sup_{x→∞} \frac{ln f(x)}{g(x)} = 1.

We prove first:

Proposition B1. If ln f(x) ∈ SR_{ρ}, then ln f_{m}(x) ∈ SR_{ρ} and lim_{x→∞} ˆf̂_{m}(x) = ρ for each m ∈ N.

Proof. Suppose that ln f_{m−1} ∈ SR_{ρ}, i.e.,

\frac{x^{n}(ln f_{m−1})^{(n)}}{ln f_{m−1}(x)} → (ρ)_{n}, x → ∞.

Using properties of the class SR and Proposition 4 (see Preliminaries), we have D(ln f_{m−1}(x)) ∈ SR_{ρ−1}, i.e., f̂_{m−1}(x) = xD(ln f_{m−1}(x)) ∈ SR_{ρ}. Since ln x ∈ SR_{0}, we have ln f̂_{m−1}(x) ∈ SR_{0}. Hence

x^{n}(ln f̂_{m−1}(x))^{(n)} = o(ln f̂_{m−1}(x)), x → ∞,

and

\frac{x^{n}(ln f_{m}(x))^{(n)}}{ln f_{m}(x)} = \frac{x^{n}(ln f_{m−1}(x) + ln f̂_{m−1}(x))^{(n)}}{ln f_{m−1}(x) + ln f̂_{m−1}(x)} ∼

∼ \frac{(ρ)_{n} ln f_{m−1}(x) + o(ln f̂_{m−1}(x))}{ln f_{m−1}(x) + ln f̂_{m−1}(x)} ∼ (ρ)_{n}, x → ∞.

Therefore, ln f_{m}(x) ∈ SR_{ρ} and, analogously, f̂_{m}(x) ∈ SR_{ρ}. Also

ˆf̂_{m}(x) = \frac{x(f̂_{m}(x))′}{f̂_{m}(x)} → ρ, x → ∞;

and, since ln f_{0}(x) = ln f(x) ∈ SR_{ρ}, the proof is finished by induction.

This proposition and our former considerations show that we could apply Theorem A to f_{m}(x) for some fixed m ∈ N. We obtain:

f_{m}^{(α)}(x) = f^{(m+α)}(x) ∼ f_{m}(x) · (f̂_{m}(x))^{α}l(f̂_{m}(x)), α < 0, x → ∞.  (B.1)

But (Proposition 4),

f̂_{k}(x) = f̂(x) + ∑_{1}^{k} ˆ f̂_{l−1}(x) = f̂(x) + O(kρ) ∼ f̂(x), x → ∞, k = 1, 2,…, m;

and

f_{m}(x) = f(x) ∏_{1}^{m} f̂_{l−1} = f(x) ∏_{1}^{m}(f̂(x) + O(1)) ∼ f(x)(f̂(x))^{m}, x → ∞.

From (B.1) and Proposition 7 it follows that:

f^{(m+α)}(x)∼f(x)·(f̂(x))^{m+α}l(f̂(x))∼ρ^{m+α}[(ln f(x))^{m+α}l(ln f(x))], α<0, x→∞.

Putting m + α = β we see that Theorem B is valid for β < m. Since m is an arbitrary integer, the proof is done.

Finally, for an illustration of our results, we give two characteristic examples. Example 1. Consider an entire function g of integer order p in the form:

g(x) := exp P_{p}(x) = ∑_{k} a_{k}x^{k},

where P_{p}(x) := b_{p}x^{p} + ⋯, b_{p} > 0, is a polynomial with nonnegative coefficients. Since ln g(x) = P_{p}(x) ∈ SR_{p}, applying Theorem B we obtain:

Proposition 8. e^{−P_{p}(x)} ∑_{k} c_{k}a_{k}x^{k} ∼ (p b_{p})^{β}c_{[x^{p}]}, x → ∞, for any r.v.s. (c_{k}) of index β ∈ R.

Example 2. Let h(x) := ∑_{k} b_{k}x^{k}, h(0) = 1 be an entire function of order ρ, 0 < ρ < 1 with negative zeros only. According to Hadamard’s Factorization Theorem, we have the representation

h(x) = ∏_{k} (1 + \frac{x}{r_{k}} ), ∑_{k} \frac{1}{r_{k}} < ∞;

where {−r_{k}} are zeros of h(x) in decreasing order.

Denoting by n(x) zero-counting function of h, we get

ĥ(x) = xD(ln h(x)) = ∑_{k} \frac{x}{x + r_{k}} = ∫_{0}^{∞} \frac{x}{x + t}dn(t),

and

xD(ĥ(x)) = ∫_{0}^{∞} \frac{xt}{(x + t)^{2}} dn(t) < ∫_{0}^{∞} \frac{x}{x + t}dn(t) = ĥ(x),

i.e., ˆ ĥ(x) < 1. So, we can apply Theorem A’ to h(x).

There is more if we notice that the zeros of h(x) are separated by the zeros of h′(x); hence, all zeros of h_{1}(x)/x are negative and, by induction, the same is valid for h_{n}(x)/x, n ∈ N. Therefore,

ˆĥ_{n}(x) < ; \begin{aligned}& \\ ( & \frac{h_{n}(x)}{x} & ) \end{aligned} < 1

and, reproducing the proof of Theorem B, we come to:

Proposition 9. If h(x) is defined as before then, without any condition,

∑_{k} c_{k}b_{k}x^{k} ∼ c_{[ĥ(x)]} h(x), x → ∞,

for r.v.s. (c_{k}) of arbitrary index.

More precisely, supposing the regular distribution of zeros, we get:

Proposition 10. If n(x) ∈ R_{ρ} then:

∑_{k} c_{k}b_{k}x^{k} ∼ (\frac{πρ}{sin πρ})^{β}c_{[n(x)]} h(x), x → ∞;

for any regularly varying sequence (c_{k}) of index β ∈ R.

Proof. As we already showed,

ĥ(x) = x ∫_{0}^{∞} \frac{dn(t)}{x + t}.

Karamata’s Tauberian Theorem for the Stieltjes transform [1, p. 40] gives: For 0 < ρ ≤ 1; n(x) ∼ x^{ρ}l(x), x → ∞ if and only if

∫_{0}^{∞} \frac{dn(t)}{x + t} ∼ Γ(1 − ρ) Γ(1 + ρ)x^{ρ−1} l(x), x → ∞.

Hence n(x) ∈ R_{ρ} implies ĥ(x) ∼ \frac{πρ}{sin πρ} n(x), x → ∞.

The rest is Proposition 9.

References

[1] N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation, Cambridge University Press, 1989

[2] R. Bojanić, E. Seneta, A unified theory of regularly varying sequences, Math. Z. 134 (1973), 91–106.

[3] E. Seneta, Regularly Varying Functions, Springer-Verlag, 1976.

[4] S. Simić, Asymptotic behavior of some complex sequences, Publ. Inst. Math. (Beograd) 39(53) (1986), 119–128.

[5] S. Simić, Polynomials and regularly varying sequences, J.Math. Anal. Appl. (to appear).

[6] E.C. Titchmarsh, Theory of Functions, Oxford, 1939.

[7] A.I. Markusevich, Theory of Analytic Functions, Moscow, 1968.

[8] G. Polya, G. Szego, Aufgaben und lehrsatze aus der Analysis, Springer-Verlag, 1964.

Matematički institut (Received 27 10 1998) Kneza Mihaila 35 11001 Beograd, p.p. 367 Yugoslavia

   ssimic@mi.sanu.ac.yu