Tropical Q-difference Analogues Of The Logarithmic Derivative Lemma,Clunie's Lemma,Valiron-Mohon'ko's Lemma And Mohon'ko's Lemma

Tropical Nevanlinna theory describes value distribution of continuous piecewise linear functions of a real variable with arbitrary real slopes,called tropical meromorphic functions.In this paper,we shall consider tropical meromorphic functions endowing with doubling condition.And we present tropical q-difference analogues of the loga-rithmic derivative lemma,Clunie's lemma,Valiron-Mohon'ko's lemma and Mohon'ko's lemma reminiscent of the corresponding results in the classical Nevanlinna theory.The main results are as follows:Theorem 1.2.1.Let f be a doubling tropical meromorphic function and q ?R\{0}.Thenm(r,f(x(?)q)(?)f(x))=o(T(r,f))holds on a set of logarithmic density 1.Corollary 1.2.3.Let f be a doubling tropical meromorphic solution of?(x,f)(?)P(x,f)=Q(x,f),where ?(x,f),P(x,f)and Q(x,f)are doubling tropical q-difference polynomials in f(x)with small coefficients.If deg(Q)?deg(?)and deg(P)?0 then for all sufficiently large r,m(r,P(x,f))=o(T(r,f))holds on a set of logarithmic density 1.Theorem 1.2.4.Given a doubling tropical q-diffierence polynomialP(x,f)=(?)??A a?(s)(?)((?)j=0 p f(x(?)qj)(?)?j)with small coefficients and of total degree deg(P)in f and its shifts,then for all suffi-ciently large r,m(r,P(x,f))=deg(P)m(r,f)+o(T(r,f))holds on a set of logarithmic density 1.Corollary 1.2.6.Let f be a doubling meromorphic solution of a doubling tropical q-difierence polynomial equation with small coechicientsP(x,?)=(?)??Aa(x)(?)((?)j=0 p ?(x(?)qj)(?)?j)=0,such that ||?||?0 for all ??A,then for all sufficiently large r and all a ? R,m(r,1?(?)(f(?)a))=o(T(r,f))holds on a set of logarithmic density 1.The structure of this paper is as follows:In chapter 1,we introduce the background of the research and the main problems solved.In chapter 2,we introduce the logarithmic derivative lemma,Clunie's lemma,Valiron-Mohon ko s lemma and Mohon'ko's lemma in the classical Nevanlinna theo-ry.In chapter 3,we introduce tropical analogues of the logarithmic derivative lemma,Clunie's lemma,Valiron-Mohon'ko;s lemma and Mohon'ko's lemma.In chapter 4,we present tropical q-difierence analogues of the logarithmic derivative lemma,Clunie's lemma,Valiron-Mohon'ko's lemma and Mohon'ko's lemma.