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Results & Lemmas (4)
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Theorem 1.
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Theorem 1. Suppose that f is transcendental and meromorphic in the plane, and that F=ff"-af'2,aeC. (i) If a^ 1 and l/(a— 1) is not a…
Theorem 1. Suppose that f is transcendental and meromorphic in the plane, and that F=ff"-af'2,aeC. (i) If a^ 1 and l/(a— 1) is not a positive integer and N(r) = o(nr,f'/f)) and I H £ | ^ < 2 (1.3)
Theorem 2.
Theorem 2. Suppose that H is transcendental and meromorphic in the plane of order p(H) such that l o I ^ ^ " ), (, 4.. 1 V, ~, _ v. v, -, „…
Theorem 2. Suppose that H is transcendental and meromorphic in the plane of order p(H) such that l o I ^ ^ " ) , (, 4 . . 1 V , ~ , _ v . v , - , „ - . „ i Ogi o g r --' and suppose further that q is a constant with q<p{H). Then H has fixpoints z with \z\ arbitrarily large and \H'(z)\>\z\q. We make some remarks about Theorem 2. First, we are free to assume that p(H) is positive, by the results of Shea and Eremenko already cited. Further, the first inequality of (1.4) implies that H has infinitel
Lemma 1.
Lemma 1. There is a positive constant cx with the following property. Suppose that c > 0 and L>(l+c)s>l+c and F(z) is analytic in the…
Lemma 1. There is a positive constant cx with the following property. Suppose that c > 0 and L>(l+c)s>l+c and F(z) is analytic in the closed rectangular region Q given by |Re(z)|^37iL, Ls^Im(z)^LSg2Ls, with |F(z)|gcL"2 there. Then the equation F(z))w(z) = 0 (2.1) has solutions U(z), V(z) in Q such that, with \EJ\ C/(z) = e-'z(l+e1(z))) C/'(z)=-ie-iz(l+e2(z)), V(z) = e(2( 1 + e3(z)), V'(z) = ie'z( 1 + e4(z)). (2.2)
Theorem 3.
Theorem 3. Suppose that 0<m<co and f is meromorphic in the plane, and that N(r) counts the zeros of ff" which are not multiple zeros of f.…
Theorem 3. Suppose that 0<m<co and f is meromorphic in the plane, and that N(r) counts the zeros of ff" which are not multiple zeros of f. If N(r) = 0(rm), then log T{r, f'lf) = O(r"), the positive constant n depending only on m. REFERENCES 1. J. M. ANDERSON, I. N. BAKER and J. CLUNIE, The distribution of values of certain entire and meromorphic functions, Math. Z. 178 (1981), 509-525. 2. W. BERGWEILER, On the zeros of certain homogeneous differential polynomials, Arch. Math. (Basel) 64 (1995),