整函数类以及它的特殊的两个子类（即复系数多项式类和指数多项式类）一直是复分析领域重点研究的对象。虽然对这几类函数的研究已取得了丰硕的成果、形成了一套较为完整的理论体系，但也还有许多基本而又十分有趣的问题没有彻底解决。例如，关于常系数指数多项式的Shapiro猜测、关于两个最大模函数相同的整函数之间到底会具有何种微妙的关系、关于多项式的Smale 平均值问题--都还没有彻底解决。本项目拟将对上面提到的三个问题进行深入研究，并争取取得实质性的进展乃至解决其中的一些问题。

The class of entire functions and its special two sub-class (i.e., the class of polynomials with complex number coefficients and the class of exponential polynomials) are always as importantly objects in the field of complex analysis. Although their theory of these function classes is already compare complete, but still has many interest open problems. For example, Shapiro’s conjecture concerning exponential polyno mials, Smale mean value problem concerning polynomials, and how delicate relation-ship between two entire functions with same maximum modulus function. In this research project, we will deeply investigate above three open problems, we strive for get essence progress even partly solve some of these three problems.