On The Zeros Of A Certain Class Of Differential Polynomials And Related Questions

In 1920s.the uniqueness theory of meromorphic functions is established by R.Nevanlinna.The Nevanlinna theory is the basis of modern meormorphic function theory,and it has great effect on the development of mathematical branches.Moreover,the Nevanlinna theory provides a useful method to the research of differential equations.Difference counterparts of Nevanlinna theory have just been built in recent years.In this paper,we research the existence of a certain class of differential equations by using the value distribution theory as an important tool.The whole paper divides into for chapters.In the first chapter,we briefly introduce some fundamental results,usual notations and main concepts in Nevanlinna value distribution theory.In the second chapter,we use the method of logarithmic derivative method and Riccati equation and so on,the nonlinear equation f3+q(z)f(z+1)=c sin(bz) was extended to more general form f3+q(z)f(z)=p1e?1z+p2e?2z,and f3(z)+q3(z)?3f(z)+q2?2f(z)+q1?f(z)=?1e?z+?2e-?z+p(z),f3(z)+p1(z)f(k)(z+d1)+p2(z)f(z+d2)=?1e?z+?2e-?zThe third chapter,we study differential polynomial form of Hayman conjecture,using the logarithmic derivative method,the method to construct auxiliary function,etc.,so improving the result of Laine and Yang,Also promoting the result of Liu and put down n ? 7 to n ? 2.The fourth chapter,we construct the auxiliary function,the first fundamental theorem,the second fundamental theorem of method and theoretical basis,for the further research of Haylan guess,at the same time it is concluded that the more beautiful results.