Radial distribution and deficiencies of the values of a meromorphic function

THEOREM 1. Let f(z) be a meromorphic function with positive zeros and negative poles. Received April 25, 1960. 1 The research of this author was supported by the United States Air Force, Contract No. AF49(638)-571, monitored by the Office of Scientific Research. 2 The research of this author was supported by a grant from the National Science Foundation (G 5253). 135

COROLLARY 1.1. The assumptions of Theorem 1 imply δ(o,/)^-A-, δ(co,/)^_4-^-. If the condition (3) is omitted, but 0 <a^limmf Mε>ίL^ i i m s u p we still have

COROLLARY 1.2. Let f(z) be an entire function with real zeros. If and if (7) Σ

THEOREM 2. Let f(z) be entire. Assume that all its zeros aβ lie on the radii defined by re iω°, re iωi, re iω™ (r > 0) , where the ω's are real. Then, there exists a positive constant K, depending only on the ω's, and such that the condition (9) Σ _ l = + co, and the condition

LEMMA 1. Let f(z) be meromorphic with zeros {αμ} and poles {6V}. Assume (1.1) ! arg αμ | :g γ < •£ (μ = l, 2 , 3, . . . ) ; Li (1.2) | a r g 6 v - π | ^ γ < | - (v = 1,2,8, •••); 3 A direct study of the lower order of our functions will be found in [2]. For the functions in Theorem 1 and its Corollaries, this study yields "best possible" bounds for μ.

LEMMA 2. Let g(z) be an absolutely convergent product of primary factors of genus 2. Assume that the zeros of g(z) lie in the sector Δ{έ) defined by (2.1) I arg z I ^ -|- - ε b »< sf.)

LEMMA 3. Let f(z) be a meromorphic function of genus not greater than 2. Assume (i) that its zeros {aμ} lie in the region Δ(s) defined by (2.1); (ii) that its poles {6V} lie in the region J*(ε) defined by I arg z — π | ^ — — ε 6 (iii) Σ f(re iβ) 1 f (JΓ/3) + (8/2) (2.7) -L\

LEMMA 4. //, in Lemma 3, we restrict the value of the para- meter ε by the inequalities (2.13) — — < ε < — , 10 6 " ~ 6 then, for all sufficiently large values of r, (2.14) Γ(r, /) ^ (1 + A){N(T9 -ί) + N(r, /)} , where A(> 0) is an absolute constant. The inequality

LEMMA 5. Let f(z) be meromorphic. Assume that there exists an integer q(^ 1) such that (3.1) (3.2) = + h \ q+1 < + Let p be an odd integer (3.3) 1 ^ p ^ q . Consider the sectors Δ^ defined by (3.4) 2πk P

Corollary 1.1 follows trivially from the inequalities (4) and (5) and the definition of deficiency.

Corollary 1.2 is contained in the following.

LEMMA 6. Let f(z) be entire. Modify the assumptions of Lemma 5 by: (i) omitting all reference to poles; (ii) omitting the restriction that p be odd (p may be any integer satisfying the inequality (3.3)). Then (3.6) still holds. The proof of Lemma 5 also yields Lemma 6 provided the integer s (even or odd) is defined by s ^ < s + l , p instead of (3.7). The definitions (3.8) remain unchanged and (3.9) takes the sharper form I ^ Q < 21 .

Theorem 2 is an immediate consequence. REFERENCES 1. A. Edrei and W. H. J. Fuchs, On the growth of meromorphic functions with several deficient values, Trans. Amer. Math. Soc. 93 (1959), 292-328. 2. and , On the maximum number of deficient values of certain classes of functions, Air Force Technical Note, AFOSR TN 60-402 (April 1960). 3. R. Nevanlinna, Eindeutige Analytische Funktionen, 2nd ed., Berlin, 1953. 4. E. C. Titchmarsh, The theory of functions 2d ed., Oxford, 1950. 5. H. Weyl, Uber die