Generalization of P\'olya's zero distribution theory for exponential polynomials, plus sharp results for asymptotic growth

An exponential polynomial of order $q$ is an entire function of the form $$ f(z)=P_1(z)e^{Q_1(z)}+\cdots +P_k(z)e^{Q_k(z)}, $$ where the coefficients $P_j(z),Q_j(z)$ are polynomials in $z$ such that $$ \max\{\deg{Q_j}\}=q. $$ In 1977 Steinmetz proved that the zeros of $f$ lying outside of finitely many logarithmic strips around so called critical rays have exponent of convergence $\leq q-1$. This result does not say nothing about the zero distribution of $f$ in each individual logarithmic strip. Here, it is shown that the asymptotic growth of the non-integrated counting function of zeros of $f$ is asymptotically comparable to $r^q$ in each logarithmic strip. The result generalizes the first order results by P\'olya and Schwengeler from the 1920's, and it shows, among other things, that the critical rays of $f$ are precisely the Borel directions of order $q$ of $f$. The error terms in the asymptotic equations for $T(r,f)$ and $N(r,1/f)$ originally due to Steinmetz are also improved.


Background on exponential sums
An exponential sum is an entire function of the form f (z) = P 0 (z)e w 0 z + · · · + P n (z)e wmz , (1.1) where the coefficients P j (z) are polynomials such that P j (z) ≡ 0 for j = 1, . . . , m. If P 0 (z) ≡ 0, we set w 0 = 0. The constants w j ∈ C are pairwise distinct, and they are called the leading coefficients of f . Thus w j = 0 for j = 1, . . . , m.
Every exponential sum (or more generally every exponential polynomial, see (1.7) below) satisfies a linear differential equation with polynomial coefficients [24]. In the literature, exponential sums can also be found as solutions to differential-difference equations [2,21], and even to linear differential equations of infinite order [2].
The functions sin z = 1 2i (e iz − e −iz ) and cos z = 1 2 (e iz + e −iz ) are typical examples of exponential sums, their quotient being tan z. In 1929 Ritt [19] showed that if the ratio g of two exponential sums with constant coefficients is entire, then g is an exponential sum also. The proof of this result was simplified by Lax in 1949 [14]. In 1958 Shapiro [21] conjectured that if two exponential sums with polynomial coefficients have infinitely many zeros in common, then they are both multiples of some third exponential sum. This claim is known as Shapiro's conjecture, and it remains unsolved.
We note that distribution of a-points and zero distribution of exponential sums are essentially the same thing because f is an exponential sum if and only if f − a is. Zero distribution of functions f of the form (1.1) has been investigated for roughly a century by several authors, starting from the case where the coefficients P j are constants and the leading coefficients w j are real and commensurable. The latter means that w j = αp j , where α ∈ R and p j ∈ Z. The early developments, with references, are surveyed in [13].
Zero distribution in the general case is based on the work of Pólya [17,18] and Schwengeler [20]. We summarize these findings in Theorem A below, for which Dickson [3] has given a generalization in the sense that the coefficients are asymptotically polynomials. Before stating the actual result, we need to agree on the notation.
Thus, if P 0 (z) ≡ 0, then W f = W 0 f . For any finite set G ⊂ C, let C(co(G)) denote the circumference of the convex hull co(G). Around a given ray arg(z) = ϕ, we define a logarithmic strip Λ(ϕ, c) = re iθ : r > 1, |θ − ϕ| < c log r r , (1.2) where c > 0. Finally, we let θ ⊥ denote the argument of a ray parallel to one of the outer normals of co (W f ). Thus the ray arg(z) = θ ⊥ is orthogonal to a line passing through one of the sides of the polygon co (W f ).
Theorem A ( [3,17,18,20]) Let f be an exponential sum of the form (1.1). Then f has at most finitely many zeros outside of the finitely many domains Λ(θ ⊥ , c). Moreover, let w j , w k be two consecutive vertex points of co (W f ), and let θ ⊥ 0 denote the argument of the ray parallel to the outer normal of co (W f ) corresponding to its side [w j , w k ]. Then the number n(r, Λ) of zeros of f in Λ(θ ⊥ 0 , c) ∩ {|z| < r} satisfies By summing over all sides of co(W f ), we get |w j − w k | = C(co(W f )). Hence Theorem A yields the asymptotic equations where r → ∞ without an exceptional set. Finally, Wittich [25] has proved the asymptotic equation Exponential sums can be generalized to exponential polynomials, which are higher order entire functions of the form where P j (z), Q j (z) are polynomials for 0 ≤ j ≤ n . A polynomial can be considered as a special case of an exponential polynomial. We assume that the polynomials Q j (z) are pairwise different and normalized such that Q j (0) = 0. This convention forces the functions g j (z) = P j (z)e Q j (z) to be linearly independent. Let q = max{deg(Q j )} denote the order of f . As observed in [22], we may then write f in the normalized form where the functions H 0 (z), H 1 (z), . . . , H m (z) are nonvanishing exponential polynomials of order ≤ q − p for some 1 ≤ p ≤ q, and m ≤ n. The leading coefficients w j (0 ≤ j ≤ m) are pairwise distinct and nonzero, except possibly w 0 . Thus H 0 (z) ≡ 0 precisely when q = deg(Q j ) holds for every j. Steinmetz [22] has proved analogues of (1.5) and (1.6) for exponential polynomials. We will improve the error terms in these asymptotic equations. So far there has been no attempt on finding a higher order analogue of (1.3). The main focus of this paper is to find one. Then a higher order analogue of (1.4) will be found in two different ways as simple consequences.
The main results are stated and further motivated in Sections 2-3. The proofs of the main results are given in Sections 4-8.

Zero distribution
So far there has been no attempts to prove a higher order analogue of Theorem A. The major part of this paper focuses in proving such a result. Before stating the actual result, we need to recall a few concepts.
If f is an exponential polynomial of the form (1.8), then its Phragmén-Lindelöf indicator function see [10, p. 993]. The indicator h f (θ) is 2π-periodic and differentiable everywhere except for cusps θ * at which for a pair of indices j, k such that j = k. These cusps are called the critical angles for f , and the corresponding rays are called critical rays for f [9,22]. An alternative way to find the critical angles is by means of the convex hull co(W f ) of the conjugated leading coefficients of f . Namely, let arg(z) = θ ⊥ j,k be the ray that is parallel to the outer normal of co(W f ) determined by two successive vertex points w j , w k of co(W f ). Then the rays arg(z) = (θ ⊥ j,k + 2πn)/q, n ∈ Z, (2.2) are critical rays for f , see [9,Lemma 3.2]. Thus, if co(W f ) has s sides/vertices, then f has sq critical rays all together. In the particular case when q = 1 the critical rays and the orthogonal rays are one and the same. Let f be given in the normalized form (1.8), where we suppose that ρ(H j ) ≤ q − p for 1 ≤ p ≤ q. The case p = 1 corresponds to the standard case, while the identity p = q reduces the coefficients H j (z) to be polynomials. In [9] it is proved that most of the zeros of f are inside of modified logarithmic strips where c > 0 and arg(z) = θ * represents one of the critical rays for f . For p ≥ 2 these domains approach the corresponding critical rays asymptotically, which does not happen in the standard case p = 1, see [9, Section 2]. More precisely, let N Λp (r) denote the integrated counting function of the zeros of f in |z| ≤ r lying outside of the union of the finitely many domains Λ(θ * , c). Then 3) The case p = 1 was proved earlier in [22], while partial results for the general case were proved in [5,16]. For (2.3) to hold, the constant c > 0 needs to be large enough (depending on f ). In what follows, we will assume that this is always the case without further notice whenever discussing the domains Λ p (θ * , c). The discussion above says nothing about the zero distribution in each individual modified logarithmic strip Λ p . This motivates us to state and prove the following generalization of Theorem A. Theorem 2.1 Let f be an exponential polynomial of the form (1.8), where the coefficients H j (z) are exponential polynomials of growth ρ(H j ) ≤ q − p for some integer 1 ≤ p ≤ q. Then the number n λp (r) of zeros of f outside of the finitely many domains Λ p (θ * , c) satisfies n λp (r) = O r q−p + log r .
Moreover, let w j , w k be two consecutive vertex points of co (W f ), and let θ ⊥ 0 denote the argument of the ray parallel to the outer normal of co (W f ) corresponding to its side [w j , w k ]. Let ε > 0, and let θ * be any critical angle for f associated with the angle θ ⊥ 0 by means of (2.2). Then the number of zeros n(r, We make some comments regarding Theorem 2.1, and discuss some immediate consequences of it. First, according to (2.1), at least two leading coefficients are needed to induce a critical ray. The example f (z) = n j=0 e jz , n ≥ 2, (2.5) x y HITW Schwengeler Pólya Figure 1: Assuming that the positive real axis is a critical ray for f , the zeros of f have been considered in different regions as follows: Pólya in sectors | arg(z)| < ε, Schwengeler in logarithmic strips | arg(z)| ≤ c log |z| |z| , and Heittokangas-Ishizaki-Tohge-Wen in modified logarithmic strips | arg(z)| ≤ c log |z| |z| p . Here ε > 0 and c > 0 are real, and p ≥ 1 is an integer.
having critical rays at arg(z) = ±π/2 shows that more than two leading coefficients can induce the same critical ray. It follows that the conjugate of any leading coefficient inducing a critical ray must be a boundary point of co (W f ). Indeed, if (2.1) holds, then From (2.2) we get that arg(z) = qθ * is an orthogonal ray for a line through the points w j and w k . Hence this line must be a support line for co(W f ), and so w j and w k are boundary points of co(W f ). However, being a boundary point of co (W f ) is not enough. In proving Theorem 2.1 it is necessary that in between two consecutive Λ p -domains precisely one exponential term is dominant. This property is explained in more detail in Lemma 7.2 below. Only those exponential terms that are associated with vertex points of co (W f ) have this property. For the function f in (2.5), the dominant term in the right half-plane is e nz , while the dominant term in the left half-plane is 1. Second, keeping in mind that w j , w k are two consecutive vertex points of co (W f ), we get |w j −w k | = C(co(W f )), where the summation is taken over all sides of co(W f ). From (2.2) we see that the orthogonal ray associated with the pair w j , w k induces q critical rays. This gives raise to the asymptotic equation Third, since a domain Λ p (θ * , c) for any p is essentially contained in an arbitrary εangle {z : | arg(z) − θ * | < ε}, it follows that the critical rays of f are precisely the Borel directions of order q of f . Note that f has no Borel directions of order ρ ∈ (q − p, q). In the case q = 1 the critical rays of f (which are the same as the orthogonal rays of f ) are precisely the Julia directions for f in a strong sense. Indeed, for any a ∈ C, f has at most finitely many a-points outside of the domains Λ 1 (θ * , c) by Theorem A.
Finally, it seems that the best possible error term in (2.4) might be O (r q−p ), but our method does not give this.
3 Asymptotic growth of T (r, f ) and N (r, 1/f ) For an exponential polynomial f of the normalized form (1.8), Steinmetz [22] has proved the asymptotic equations These are higher order analogues of (1.5) and (1.6). The next result improves the error term in (3.2) even in the standard case p = 1. It is obvious that the logarithmic term in (3.3) below is only needed when q = p, that is, when the coefficients H j (z) are ordinary polynomials.
Let f be the exponential polynomial in (1.7), and suppose that f can be written in the normalized form (1.8), where ρ(H j ) ≤ q − p for 1 ≤ p ≤ q. If one of the polynomials Q j (z) vanishes identically, we may suppose that it is Q 0 (z), and that no other polynomial Q j (z) vanishes identically. In this case H 0 (z) reduces to an ordinary polynomial in z. Under these conditions, we have the following result that improves the error term in (3.1).
We conclude this section by the following quick consequence of (3.1). This is a weaker form than (2.6), but it is needed in proving Theorem 2.1, whereas (2.6) is a consequence of Theorem 2.1.
Similarly, using log x ≤ x − 1 for x ≥ 1, we get Choose δ(r) = ε(r), and all error terms above are of the form o (r q ). ✷

Lemmas for Theorem 3.1
Let f be an exponential polynomial with no zeros. Then f = e P , where P is a polynomial of degree p, and there exist constants C > 0 and R > 0 such that for |z| > R. If f has finitely many zeros, then f = Qe P , where Q is also a polynomial. The conclusion in (4.1) holds in this case also, but possibly for different constants C and R.
For an exponential polynomial f with infinitely many zeros, a lower bound for log |f (z)| is given in [9, Lemma 5.1], which originates from [23,Theorem V. 19]. This estimate is valid for all z outside of certain discs centered at the zeros of f . In our applications, however, we need to be more flexible with the size of these discs, as is described next. Lemma 4.1 Let f be an entire function of order ρ = ρ(f ), let λ = λ(f ) denote the exponent of convergence of zeros of f , and let k : (0, ∞) → [1, ∞) be any non-decreasing function. Suppose that λ ≥ 1 and that the number of zeros of f in |z| < r satisfies whenever z = re iθ lies outside of the discs |z − z n | ≤ 1/k(|z n |) and |z| ≤ e.
Proof. We may factorize the entire f as f = Qe P , where P is a polynomial and Q is a canonical product formed with the zeros z n of f . Set p = deg(P ), in which case ρ = max{p, λ}, and q = ⌊λ⌋. Since the integral Hence q is the genus of the sequence {z n }, see [23, p. 216] for the definition. This allows us to write Q in the form We also write Taking positive real parts on both sides of the above equation results in where r n = |z n |. Similarly as in the proof of [9, Lemma 5.1], we obtain Σ 2 = O (r q ). This estimate has nothing to do with the size of the discs around the points z n . In Σ 1 , we have If z lies outside of the discs |z − z n | ≤ 1/k(|z n |) and |z| ≤ e, it follows that log + z n z − z n ≤ log (r n k(r n )) ≤ log (2rk(2r)) , because r n ≤ 2r and k is non-decreasing. Thus Σ 1 ≤ n(2r) log (2rk(2r)) + 2 q r q 2r e dn(t) t q + O(1) ≤ O r λ log (rk(2r)) + 2 q qr q 2r e n(t) t q+1 dt = O r λ log (rk(2r)) .
The reasoning above shows that there exists a constant B > 0 such that log |Q(z)| ≥ −Br λ log (rk(2r)) whenever z = re iθ lies outside of the aforementioned closed discs. Since P is a polynomial of degree p, there is a constant C > 0 such that log e P (z) ≥ −Cr p for |z| > e. The assertion now follows by choosing A = B + C. ✷ whenever z = re iθ lies outside of the discs |z − z n | ≤ 1/k(|z n |) and |z| ≤ e.
Remark. We may choose k(x) = x λ+ε , ε > 0, in Lemma 4.1 and k(x) = x q log 2 x in Corollary 4.2. For these choices of k, it follows by Riemann-Stieltjes integration that Thus the projection I of the discs |z − z n | ≤ 1/k(|z n |) on the positive real axis has finite linear measure.
The following lemma follows implicitly from the discussions in [11,Section 3]. For the convenience of the reader, we give a direct proof. Lemma 4.3 Suppose that T : R + → R + is eventually a non-decreasing function. Suppose further that there exist constants C > 0, q > 0, 0 ≤ p < q and s ≥ 0 such that, as r → ∞, T can be written in the form where E ⊂ [1, ∞) has finite logarithmic measure (or finite linear measure). Then the asymptotic equality in (4.3) holds for all r large enough.
Proof. Let ε ∈ (0, min{1, C}). Then there exist constants R 1 = R 1 (ε) > 1 and C 1 > 0 such that T (r) is increasing for r > R 1 and (C − ε)r q − C 1 r p log s r ≤ T (r) ≤ (C + ε)r q + C 1 r p log s r, where r ∈ (R 1 , ∞) \ E. Moreover, due to p < q we may suppose that R 1 is chosen large enough so that the lower bound and the upper bound for T (r) are increasing functions of r. Then, using [7, Lemma 5], we can find an R 2 = R 2 (ε) > max{2, R 1 } such that, for r > R 2 , and similarly where R 2 = R 2 (ε ′ ) > 2 and C 2 = 2 p+s C 1 . This yields the assertion. ✷ For a given c > 0 there exists an R 0 > 0, with the following property: If z = re iθ ∈ Λ p (θ * , c) for any θ * and any r ≥ R 0 , and if h f (θ) = ψ j (θ), then

Proof of Theorem 3.1
We divide the proof into four steps for clarity.

Preliminaries.
Let arg(z) = θ j , j = 1, . . . , k, be the critical rays of f organized such that 0 ≤ θ 1 < θ 2 < · · · < θ k < 2π. Set θ k+1 = θ 1 + 2π. For j = 1, . . . , k, define F j = (θ j , θ j+1 ) and where c > 0 is sufficiently large. The domain Λ p (θ j , c) curves asymptotically towards the critical ray arg(z) = θ j , provided that p ≥ 2. The interval F j (r) corresponds to the arguments in between two consequtive domains Λ p (θ j , c) and Λ p (θ j+1 , c). Clearly F j (r) → (θ j , θ j+1 ) = F j as r → ∞ for any p ≥ 1. Since f has no poles, we have For j = 0, . . . , m, we define By Lemma 4.4, each set E j (r) is a finite union of intervals F i (r). Thus set E j is a finite union of intervals F i . If E 0 = ∅, then P 0 (z) ≡ 0 and w 0 = 0, so the definition of the set E j implies that the terms |H j (z)e w j z q |, j = 1, . . . , m, are bounded for arg(z) = θ ∈ E 0 . On the other hand, if θ ∈ E j for some j, then |H j (z)e w j z q | ≤ M(r, H j ). Keeping in mind that ρ(H j ) ≤ q − p, we conclude in both cases E 0 = ∅ and E 0 = ∅ that T (r, f ). To estimate the integrals in (5.1) over the sets E j , we write

Upper bound for
If z = re iθ and θ ∈ E j , it follows that where k f (θ) is the support function for the convex set co(W 0 f ) [15, p. 74]. Therefore where the last identity follows by Cauchy's formula for convex curves. This formula can be found in many books on integral geometry.

Lower bound for
T (r, f ). If q = 1 or if p = q, then the coefficients H j (z) are polynomials, and no exceptional set other than |z| is large enough occurs when estimating T (r, f ) downwards. Hence we suppose that 1 ≤ p ≤ q − 1. Observe that for all x ≥ 0 and y > 0. So, if z = re iθ and θ ∈ E j (r), by applying this observation to (5.2), we have The functions H j (z) are of order ≤ q − p ≤ q − 1 and of finite type. Therefore there exists a constant A > 0 such that log M(r, H j ) ≤ Ar q−p log r and log |H j (z)| ≥ −Ar q−p log r.
The latter inequality is valid for all |z| = r outside of a set I ⊂ (0, ∞) of finite linear measure. This follows by applying Lemma 4.1 with k(x) = x q to each of the coefficients H j (z), and then taking the union of all the exceptional sets involved. See also the remark following Lemma 4.1. We note that k(x) could be chosen to have less growth at this point, but the particular growth rate fixed here is needed later on. Choosing c > 2a in Lemma 4.4, we have where |z| = r ∈ I. Recall that each set E j is a finite union of intervals F i . In fact,

From (5.1), (5.3) and (5.4), it then follows that
where the last identity follows again by Cauchy's formula and by the fact that h f (θ) = k f (qθ).

Proof of Therem 3.2
Let us begin with the particular case Q 0 (z) ≡ 0. As the functions g j (z) = P j (z)e Q j (z) , j = 1, . . . , n, are linearly independent, they form a fundamental solution base for a linear differential equation L n (g) := g (n) + a n−1 (z)g (n−1) + · · · + a 0 (z)g = 0, (6.1) see [12,Proposition 1.4.6]. The coefficient functions a j (z) can be expressed as quotients of Wronskian and modified Wronskian determinants of the functions g j (z). Note that This means that the exponential term e Q j (z) remains in differentiation, and further differentiations will not change that. Therefore, from each column of each determinant we can pull out the single exponential term as a common factor. Thus exp Q 1 (z) + · · ·+ Q n (z) is a common factor for each determinant, and they cancel out in the quotients. The entries of the remaining determinants are polynomials. Hence the coefficients a j (z) in (6.1) are rational functions. If P 0 (z) would be a solution of (6.1), it would have to be linearly dependent with the solutions g 1 , . . . , g n . Thus P 0 (z) would be transcendental, which is a contradiction. It follows that a n (z) := L n (P 0 ) ≡ 0 is a rational function. We conclude that f is a solution of the non-homogeneous equation f (n) + a n−1 (z)f (n−1) + · · · + a 0 (z)f = a n (z).
Dividing both sides by f a n (z) and using the lemma on the logarithmic derivative together with the fact that the coefficients are rational gives us (3.5).
Now the discussion above can be applied to g giving us m(r, 1/g) = O(log r) and, a fortiori, Suppose in addition that H 0 (z) ≡ 0. Due to the assumption ρ(H j ) ≤ q − p, we may assume, without loss of generality, that deg(Q 0 ) ≤ q − p. Thus, using Theorem 3.1 to f and using (6.2), we get We deduce the estimate (3.4) by combining (6.3) and (6.4). Finally, we suppose that H 0 (z) ≡ 0, in which case deg(Q j ) = q for every j and m = n. Let U = {w 1 − w 0 , . . . , w m − w 0 } be the set of conjugate leading coefficients of g, and set U 0 = U ∪ {0}. By a simple vector calculus, we conclude the following: If w 0 is a boundary point of co(W f ), then 0 is a boundary point of co(U), while if w 0 is an interior point of co(W f ), then 0 is an interior point of co(U). Thus, applying Theorem 3.1 to g, we get T (r, g) = C(co(U 0 )) r q 2π + O r q−p + log r = C(co(W f )) r q 2π + O r q−p + log r .
The final assertion (3.6) follows from this and (6.3).

Lemmas for Theorem 2.1
The following lemma is considered for more general curves in [4] but in Cartesian coordinates. However, our situation is very delicate, and we need the representation particularly in polar coordinates.
Lemma 7.1 Let p ≥ 1 be any real number, and let U be any collection of Euclidean discs D n = D(z n , r n ), where the center points z n ∈ C are ordered according to increasing moduli, |z n | → ∞, r n > 0, r n → 0 and |zn|>e |z n | p−1 r n log |z n | < ∞.
Then the set C ⊂ (0, ∞) of values c for which the curve ∂Λ p (0, c) = z = re iθ : r ≥ 1, | arg(z)| = c log r r p meets infinitely many discs D n has linear measure zero.
Proof. If z ∈ ∂Λ(0, c), then z = r cos c log r r p ± ir sin log r r p ∼ r ± ic log r r p .
Asymptotically this corresponds to the Cartesian curves y = ±c log x x p−1 . From this representation we see that the cases p = 1 and p > 1 have a different geometry. Nevertheless our proof works for both cases simultaneously.
By the Cartesian representation we may suppose that the points z n lie in the right half-plane in between the lines y = ±x. Since |z n | → ∞ and r n → 0, there exists a positive integer N such that |z n | − r n ≥ |z n |/2 ≥ e, n ≥ N.
Note that the function x → log x x p is strictly decreasing for x ≥ e. Moreover, it suffices to consider the portion of ∂Λ p (0, c) located in the upper half-plane, call it Λ + (c) for short.
Suppose that a disc D n lies asymptotically in between the curves Λ + (c 1 ) and Λ + (c 2 ), where c 1 < c 2 , and let ζ 1 and ζ 2 denote the respective intersection points. The disc D n can be seen from the origin at an angle 2θ n , where sin θ n = r n /|z n |.
Let ε > 0. Then there exists an N(ε) ∈ N such that ∞ n=N (ε) 2 p pπ|zn| p−1 rn log(|zn|/2) < ε. Let C ε ⊂ (0, ∞) denote the set of values c such that the curve Λ + (c) meets at least one of the discs D n , where n ≥ N(ε). Then C ε has linear measure < ε. Since C is contained in all the sets C ε , ε > 0, it follows that C has linear measure zero. ✷ Through the rest of this section, let k : (0, ∞) → [1, ∞) denote any non-decreasing function. To simplify the notation, we restrict to the growth rate k(r) = O (r σ ) for some σ > 0, even though the results that will follow would allow even faster growth. Let f be an exponential polynomial in the normalized form (1.8), and let {z n } denote the sequence of zeros of f and of all of its transcendental coefficients H j , listed according to multiplicities and ordered according to increasing moduli. Finally, let C(f, k) denote the collection of discs |z − z n | ≤ 1/k(|z n |). For simplicity, we may also assume that a disc |z| < R for a suitably large R > 0 is included in C(f, k).
The next auxiliary result is obtained by modifying the proof of [9,Theorem 2.4], see also the remark following [9,Theorem 2.4].
Lemma 7.2 Let f be an exponential polynomial in the normalized form (1.8), where we suppose that ρ(H j ) ≤ q − p for 1 ≤ p ≤ q. Then there exist constants A > q and c > 0 with the following properties: If Λ denotes the domain in between any two consecutive zero-domains Λ p (θ * 1 , c) and Λ p (θ * 2 , c) of f , and if z ∈ Λ \ C(f, k), then there exists a unique index j 0 such that f (z) = H j 0 (z)e w j 0 z q (1 + ε q−p (r)), (7.2) where |ε k (r)| ≤ B exp −Ar k log r , k ≥ 0, |z| = r, and B > 0 is some constant. Denote For the same values of z and j 0 as above, we have Proof.
(1) In order to prove (7.2), we first suppose that q ≥ 2 and that 1 ≤ p ≤ q − 1. The latter means that all coefficients H j (z) are assumed to be transcendental. There exist constants A > q and r 0 > 0 such that log M(r, H j ) ≤ Ar q−p , j ∈ {0, . . . , m}, (7.4) whenever r ≥ r 0 . We conclude by Corollary 4.2 that there exists a constant A > q such that log |H j (z)| ≥ −Ar q−p log r, j ∈ {0, . . . , m}, (7.5) whenever z ∈ C(f, k). We may suppose that the constant A in (7.5) is the same as that in (7.4) by choosing the larger of the two. Noting that r q ψ k (θ) = ℜ (w k z q ), we make use of Lemma 4.4 by choosing c > 4A a : If z = re iθ ∈ Λ and if r is large enough, then there exists a unique index j 0 such that ℜ (w j 0 z q ) ≥ ℜ (w k z q ) + 4Ar q−p log r (7.6) for all k = j 0 . If in addition z ∈ C(f, k), then the estimates (7.4)-(7.6) applied to (1.8) yield On the other hand, where z ∈ Λ \ C(f, k). This discussion covers the case w j 0 = w 0 = 0 also, that is, the case when f (z) − H 0 (z) = ε p−q (r) in λ \ C(f, k). If some (but not all) coefficients are polynomials, the previous reasoning simplifies. Indeed, if a particular coefficient H j (z) is a polynomial, then the growth of |H j (z)| is comparable to |z| deg(H j ) for r large enough. Consideration of the set C(f, k) is not needed for this particular coefficient, as we only need to assume that |z| is large enough.
Suppose then that q ≥ 1 is any integer and p = q, that is, all of the coefficients H j (z) are polynomials. This covers the remaining case. There exist constants A 1 > 0 and A 2 > 0 such that where d j = deg(H j ) and d = max{d j }. Suppose that z = re iθ ∈ Λ, and that r is large enough. Let A = max{d, 2}. We choose c > 4A a in Lemma 4.4, and find that there exists a unique index j 0 such that On the other hand, where z ∈ Λ \ C(f, k). This completes the proof of (7.2).
(2) In order to prove (7.3), we first notice that where G j (z) = H ′ j (z) + qw j z q−1 H j (z) for j ≥ 0. If G j (z) ≡ 0, then either H j (z) is a constant and w j = 0 or ρ(H j ) = q. The latter is clearly impossible, while the former is possible only in the case when j = 0. Hence f ′ is an exponential polynomial with coefficients being of order ≤ q − p, and shares the leading coefficients with f , except possibly w 0 . Thus, by considering separately the cases j 0 = 0 and j 0 = 0, the proof of Part (1) applies to f ′ , and we obtain (7.3) for the same index j 0 that appears in (7.2). This completes the proof. ✷ If any of the coefficients H j in (1.8) is transcendental, it also can be represented in an analogous normalized form as f , where the coefficients are either polynomials or exponential polynomials. Proceeding from one generation of transcendental coefficients to the next, the order of growth decreases at least by one. Obviously there are at most q − 1 generations of transcendental descendant coefficients all together.
Let {z n } denote the sequence of zeros of f , of the transcendental coefficients H j of f , and of all transcendental descendants of these coefficients, listed according to the multiplicities and ordered according to increasing moduli. Let then C 0 (f, k) denote the collection of all discs |z − z n | ≤ 1/k(|z n |), and suppose that a suitably large disc |z| < R is also included in C 0 (f, k). Clearly, C(f, k) is a sub-collection of C 0 (f, k).
Let Θ denote the set consisting of the critical angles of f in (1.8), the critical angles of the transcendental coefficients H j of f , and the critical angles of all transcendental descendants of these coefficients. Some of the critical angles of f may coincide with those of its coefficients or descendant coefficients. Nevertheless, Θ is a finite subset of [0, 2π), so the elements θ j of Θ can be ordered, say 0 ≤ θ 1 < θ 2 < · · · < θ l < 2π. Set θ l+1 = θ 1 . Lemma 7.3 Let f be an exponential polynomial in the normalized form (1.8), where we suppose that ρ(H j ) ≤ q − p for 1 ≤ p ≤ q. Let θ j , θ j+1 be two consecutive elements of Θ, and let Λ be the domain in between the domains Λ p (θ j , c) and Λ p (θ j+1 , c), where c > 0 is sufficiently large. If z ∈ Λ \ C 0 (f, k), then there exists a unique index j 0 and constants C 0 , . . . , C q−p−1 ∈ C such that f ′ (z) f (z) = qw j 0 z q−1 + C q−p−1 z q−p−1 + · · · + C 0 + O r −1 . (7.7) Proof. We will make use of the proof of Lemma 7.2 not only for f but also for the coefficients and the descendant coefficients of f . For each function the proof is used, new constants A > q and c > 0 are found. Since there are at most finitely many functions involved, we may choose A and c to be the maxima of all these corresponding coefficients. Independently on whether w j 0 = 0 or w j 0 = 0, it follows directly from (7.2) and (7.3) that f ′ (z) f (z) = qw j 0 z q−1 + H ′ j 0 (z) H j 0 (z) (1 + ε q−p (r)). (7.8) Suppose that H j 0 (z) is a polynomial. Then (7.8) yields +O exp (q − 1) log r − Ar q−p log r = qw j 0 z q−1 + O r −1 , z ∈ Λ \ C 0 (f, k), because of A > q. This proves (7.7) in the case when H j 0 (z) is a polynomial. Suppose next that H j 0 (z) is transcendental. Then q − p ≥ 1, so we may write H j 0 (z) in the normalized form with coefficients being either exponential polynomials of order ≤ q − p − 1 or ordinary polynomials. Analogously as in (7.8), the proof of Lemma 7.2, applied to H j 0 (z) instead of f , yields H ′ j 0 (z) H j 0 (z) = C q−p−1 z q−p−1 + K ′ (z) K(z) (1 + ε q−p−1 (r)), (7.9) where C q−p−1 ∈ C and where K(z) is either an exponential polynomial of order ≤ q − p − 1 or an ordinary polynomial in z.
If K(z) is transcendental, then we continue inductively in this fashion by reducing the order on each step by one. Eventually the descendant coefficient must reduce down to a polynomial, and, as such, its logarithmic derivative is of growth O (r −1 ). This gives us the representation (7.7). ✷ Finally, we remind the reader of the following well-known standard growth estimate for logarithmic derivatives by Gundersen [6].
By putting everything together, we conclude that Keeping (8.1) in mind, the left-hand side of (8.2) is the number of zeros n(r, Γ) of f in a domain bounded by the closed curve Γ that depends on r. Since counting functions are always non-negative, it follows that b j − b k = |b j − b k |. The situation would be symmetric if we would have chosen the leading coefficients w j and w k the other way around in the reasoning above. Moreover, from (2.1), we get ℜw j e iqθ * = ℜw k e iqθ * , which can be written as a j cos qθ * − b j sin qθ * = a k cos qθ * − b k sin qθ * .
Since we have chosen θ * = 0, it follows that a j = a k . Therefore, we have |w j −w k | = |b j −b k |, so that (8.2) now reads as n(r, Γ) = |w j − w k | 2π r q + O r q−p log 2+α r .
The assertion follows from this via Lemma 4.3 by writing α = 1 + ε.