【摘要】 <正> §1.引言1932年 Rogosinski 首先研究了单位圆 E:|z|<1内正则的典型实照函数,这种函数的全体成一函数族 T_r（E）假如 f（z）∈T_r（E）,那末 f（z）=z+a₂z²+…在|z|<1是正则的,且满足条件更多还原

【Abstract】 In the circular ring R_q:q<|z|<1 let the function f（z） be meromorphic andsatisfy the condition（?）The class of all such functions f（z）will be denoted by T_r（R_q）.Let the poles p_i of f（z）∈Tr（R_q）be arranged so that（?）Let the Laurent expansion for f（z）in the ring p′<|z|<P（P′=|p-1|,p=|p₁|）be（?）then the b_n’s are real numbers.Fixing p’ and p,denote the class of all such functionsf（z）by T_r（p′,p）,q<p′<p<1.All the functions f（z） of T_r（Rq） such that they are regular onp=|P-1|<|z|<<1（q<|z|<|（p₁）|=p） form a sub-class T_r（p′,1）（T_r（q,p））.The function（?）is schlicht and typically-real in R_q,where a is real with absolute value less than1/2（q+1/q）.Theorem 1.If f（z）∈T_r（R_q）,then the inequality（?）holds in R_q for any real a,（?）provided that（?）Theorem 2.（?）then for each z in R_q,（?）（z）≠O,there exists areal numbera=a（z）,|a|<1/2（q+1/q）,such that（?）Theorem 3.Under the conditions of Theorem 2,we have （?）where L_b is a curve with the extremities z=+q and z=-q,depending on the positiveparameter b （b<|S_q（z,a）|）and lying in the upper half or in the lower half of thering according as（?）respectively,such that L_b approaches a semi-circumference of |z|=q as b→O.Theorem4.Let the negative number-m_j be the residue of f（z）at the polep_j.Then（?）（i）if f（z）∈T_r（q,p）,and b₁-b_-1=1,we have（?）Theorem 5.Let b₀=0,then（?）when n is even;and（?）when n（>1）is odd,where（?）when n is even;and（?）when n（>1）is odd;（?） when n is even;（?）when n（>1）is odd.All the above estimates are precise.Theorem 6.Corresponding to a function f（z）of T_r（R_q）there exists a functiong（z）regular and typically real in R_q,such that（?）where the p_j（j=O,±1,±2,...）are the poles of f（z）in R_q.From this result we can deduce an integral representation of f（z）∈T_r（R_q）.更多还原