Title: ON THE FILLING CIRCLES WITH RESPECT TO THE MEROMORPHIC FUNCTIONS IN THE UNIT DISK
Abstract: Milloux,H.asserted in 1987:Let f(z) be a meromorphic function in the unit aisk.If its Navanlinna Characteristic function T(r.f) satisfies the following condition lim~-_(r→1)log~2(1-r)/1/T(r,f)=+∞ then there exists a Series of the filling circles in which the index tends to infinite and the sight of the unit disk under the angles tends to zero. For a meromorphic function f(z)in the unit disk,if it satisfies (T(r,f))/(log1(1-r))=+∞,(B) then we say f(z)satisfies Picard's theorem.Millou's assertion shows that Picrd's theorem is verified in the preceding filling circles. We know that this result is perfectly analogous to the known fact of the filling circles with respect to the meromorphic functions in the plane,so long as the expres- sion log1/(1-r) in(A)is replaced by log r.In the case of the plane,there is an example(as Ostrowski's example)which shows that it is impossible to obtain any better result. Milloux conjectured that if f(z)is a meromorphic function in the unit disk and merely satisfies condition(B)of Picard's theorem,then there still exists a series of the filling circles. I assert that this is true and have proved: Theorem 2.Let f(z)be a meromorphic function in the unit disk and satisfy (B),then there exists a series of the filling circles in which the index tends to infinity and the sight of the unit disk under the angles tends to zero. Theorem 3.Let f(z)be a meromorphic function of finite positive order ρ in |z|1 and ω(r)denote a positive continuous real function,which is non-increasing and is defined in the interval 0r1,and 0ω(r)1/(100),(ω(r))/(ω(r'))K'_0,where r'= r+ω(r)(1-r)and K'_0 is an absolute constant.If ρ(1/(1-r))is an accurate order of f(z),U(t)=t,~(ρ(t)),i.e.p(t)is a piecewise differentiable monotone function defined in the interval 0t_0≤t+∞ and satisfies; 1°()? ρ(t)=ρ, ρ'(t)t log t=0; 2° (T(r,f))/(U(1+(1-r)))=1, then there exists a series of the filling circles,the pseudo-radius of which is of ω(r). The non-Euclidean center of the filling circles is in the ring r|z|r'.The index of the circles is of N(r')=K_ρU(/(1-r'))ω2(r')and the sight of the unit disk under the angles is less than 8ω(r),where K_ρ is a positive constant with respect to ρ. Theorem 4.Suppose f(z),ρ,ρ(1-(1-r))are the same as in theorem 8,then there exists a series of the filling circles,the index of which is of N.The non-Euclidean center of the filling circles is in the ring r|z|r',where r'=r+ω(r)(1-r)and N is a positive constant(integral number).The pseudo-radius of the filling circles is of ω(r)={N/(U(1/(1-r)))}~(1/2)K'_ρ(as -→1), where K'_ρ is a constant with respect to p. Theorem 5.Let f(z)be a meromorphic function of infinite order in|z|1. Suppose σ(1/(1-r))is an accurate order of f(z),V(t)=t~(σ(t)),i.e.,σ(t)is a continuous non-decreasing function defined in 0t_0≤t∞ and σ(t)=+∞,satisfies the following conditions: 1° V(t+a[V(t)])e~r(as t is sufficiently large)whore r is a given positive constant and a(u)is a positive continuous non-increasing function,defined in u≥1 and integral from n=1 to ∞ a(u)/u du+∞;(log a(u))/u=0. 2°(T(r,f))/(V(1/(1-r)))=1. If q(t)is also a continuous non-decreasing function defined in t≥1 such that q(t)=+∞,q(t)≥1 and satisfies: 1°q(t[1+2(q(t))])Cp(t)(t is sufficiently large), 2° a[V(t)]q(t)/t=+∞, where C is a positive constant.Then there exists a series of the filling circles,the pseudo-radius of which is of ω(r)=1/q(1/(1-r)). The non-Euclidean center of the pseudo-circles is in the ring r|z|r',where r'=r+∞(r)(1-r). The index is of N(r')=1/H V(1-(1-r'))σ(1-(1-r'))σ(1-(1-r'))ω~2(r'), where H is a positive constant with respect to C and r.The sight of the unit disk under the angles is still less than 8ω(r). Theorem 6.Supposef(z),σ(t),a(u),V(t)are the same as in theorem 5.If a(u)satisfies the following inequality 1/t a[V(t)]{V(t)σ(t)}~(1/2)=+∞, then there exists a series of the filling circles,the index of which is of the constant N.The non-Euclidean center of the pseudo-ciroles is in the ring r|z|r',where r'=r+ω(r)(1-r).The pseudoradius is of ω(r)=H'N~(1/2){V(V(1/(1-u)))σ(1/(1-r))}~(1/2), where H' is a positive constant with respect to f and V(t). The results of theorems 3~6 are perfectly analogous to the known facts of the filling circles with respect to meromorphic functions in the plane. It is well known that there is an example which shows that it is impossible to improve condition(B)of Picard's theorem in the unit disk.Hence it is impossible to get any better result in theorem 2.
Publication Year: 1981
Publication Date: 1981-01-01
Language: en
Type: article
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