On the Zeros of Meromorphic Functions of the Form F(z) = ~-~o~ ~a K=l Z-z~

We study the zero distribution of meromorphic functions of the formf(z) = ~--]ff=l at where a t. > 0. Noting thatfts the complex conjugate of Z--Z k the gradient of a logarithmic potential, our results have application in the study of the equilibrium points of such a potential. Furthermore, answering a question of Hayman, we also show that the derivative of a meromorphic function of order at most one, minimal type has infinitely many zeros. 1. Introduct ion Cons ide r a m e r o m o r p h i c func t ion oo ~ ak a t > 0 . (1.1) f ( z ) = z zt.' k=l We suppose that (1.2) ~ ak k = l ~ < ( x ) and thus that the series in (1.1) converges abso lu te ly for all z E C , z # zk. The f u n c t i o n f is the complex con juga te to the g rad ien t o f the logar i thmic potent ia l = , z [ (1.3) u(z) = Z ak log k = 1 Z k which is a s u b h a r m o n i c func t ion of order at mos t one, conve rgence class. This fo l lows f rom (1.2). The zeros o f f are the e q u i l i b r i u m or cri t ical poin ts o f u. If the * Supported by an NSF grant. t Research carried out during a visit to the University of Illinois, funded by an NSF grant. ~; Research carried out at the University of York while serving as a British Science and Engineering Research Council (SERC) fellow. The author gratefully acknowledges the hospitality and support extended to him by the Department of Mathematics. 271 JOURNAL D'ANALYSE MATHI~MATIQUE, Vol. 62 (1994) 272 A. EREMENKO ET AL. ak are all positive integers, we may also consider the entire function 1-!5 t=l zt / In this c a s e f = F'/F. The zeros o f f were studied in [3] where the following results were obtained: T h e o r e m 1.1 I f all the at are positive integers, then f has infinitely many zeros. T h e o r e m 1.2 I f all the aj are positive real numbers and 2-, aj = o(v/r), r , oe, (1.4) {j:lzjl_a >Oin(1.1). Thenf hasaninf ini tesetof zeros. Thus Theorem 1,3 is stronger than Theorem 1.1. In [3] it was proved that if F is a meromorphic function of order less than 1/2, then F' /F has infinitely many zeros. Examples were given to show that there exist merormorphic functions of any order greater than or equal to 1/2 whose log derivatives have no zeros. This led Hayman to ask whether the derivative of a meromorphic function of order less than one has any zeros. Our next theorem answers this question affirmatively. ZEROS OF MEROMORPHIC FUNCTIONS 273 T h e o r e m 1.5 Let F be a transcendental meromorphic function of order at most one, minimal type; then F' has infinitely many zeros. For F of order at most one convergence class, it is easily seen that Theorem 1.5 is equivalent to the following T h e o r e m 1.6 If f is as in (1.1) and the ak are integers, with ak > -1 for j > jo, thenf has infinitely many zeros. To establish the equivalence, note that under the conditions of Theorem 1.6 we have f = g'/g, where g is a meromorphic function with at most finitely many multiple poles, having order at most one, convergence class. Applying Theorem 1.5 to F = 1/g we conclude t h a t f = g'/g has infinitely many zeros. Conversely if F is a meromorphic function of order at most one convergence class, then either F has infinitely many multiple zeros, or else we may apply Theorem 1.6 t o f = F ' / F . In either case Theorem 1.5 is true. We mention that Theorem 1.5 is sharp. Indeed for p > 1, it was shown in [3] that there exists an entire function G of order p such that G'/G has no zeros. Then F = 1/G is meromorph ic of order p and F ' has no zeros. The next two theorems give quantitative information on the distribution of the zeros o f f in certain cases. T h e o r e m 1.7 l f the function f defined in (1.1) has lower order )~ < 1 and (1.5) O < a < a k 0 and let g be a transcendental meromorphic function, o f order at most one, minimal type, having only finitely many poles. Let F be a path such that F(t) --* c~ as t ---, cx~ and (2.1) loglg(z)[ ~ as z ~ , z E F . log Iz[ Then there exists a domain S with the following properties: a. l f O(t) = meas{0 E [0, 27r] : te iO E S}, then for some ro > O, we have fro dt q~0(r) := log r rc tO( '---') --~ +oo a s r --+ ~ ~ . b. For some ri > 0 the part o f f lying in {z : Iz[ > ri } is contained in S. e. For any Zl, z2 E S there exists a path "~ from zi to z2 satisfying