Argument Distribution and Deficient Values

We investigate above bound of total number of deficient values of a transcendental meromorphic function and its derivatives of every order if most of its zeros and poles distribute along finitely many rays starting from the origin and prove that the bound is the number of the rays under some assumption, for example, the function considered is of finite lower order. Next we discuss relations between the numbers of deficient values and common T directions of the functions and their every order derivatives, and demonstrate that total number of deficient values of a meromorphic function and its every order primitive is not greater than the number of common T directions of the functions and its every order derivatives. We then go to the case of entire functions. For example, we verify that a bound of total number of deficient values of the function and its derivatives and primitives of all orders is two times its lower order for an entire function with the finite positive lower order. Finally, we make a simple survey of some of the celebrated related works of Edrei and Fuchs’ and Yang and Zhang’s and others.