| In this paper, we mainly research the uniqueness concerning shared function and the u-niqueness of meromorphic functions with their differential and q-difference-differential poly-nomials, meanwhile we investigate normal family of meromorphic functions sharing functions.The full text is divided into four chapters.In Chapter1, we outline some basic concepts, results and notations of Nevanlinna valuedistribution theory; the Nevanlinna Characteristic, the results of difference Navenlinna theories;normal family and normal criterions etc.In Chapter2, we first research if for the a(z) which is small function of f (z), transcen-dental meromorphic functions f (z) and g(z) satisfy E(a,[fnP (f)](k))=E(a,[gnP (g)](k)),then we get a normal conclusion, which greatly improves the conclusion of zhang, chen andFang. Furthermore we research the case that transcendental meromorphic functions f (z) sat-isfies Θ(∞, f)>4n+m, we can get the relation of f and g. Meanwhile we research thecase that transcendental meromorphic functions f (z) and g(z) satisfy E(p(z),[fnP (f)](k))=E(p(z),[gnP (g)](k)), where p(z) is a nonzero polynomial, we can get the relation of f and g.In Chapter3, We investigate the uniqueness problem of the finite order of difference op-erators of entire functions f (z) and g(z), namely fncf(z) and gncg(z) share a(z) CM, theresult greatly improves the conclusion of chen and Li. And we research the case that unique-ness problem of the zero order of q-difference-differential polynomial of transcendental entirefunctions and meromorphic functions f (z) and g(z) share one small function, namely if tran-scendental entire functions f (z) and g(z) satisfy [fn(fm1)f(qz)](k)and [gn(gm1)g(qz)](k)share the value1CM, then f=tg, tm=tn+1=1, if transcendental meromorphic functionsf(z) and g(z) satisfy the above condition, we have fn(fm1)f(qz)=gn(gm1)g(qz).In Chapter4, We mainly investigate some normal criterions of meromorphic family F, iffor (f, g)∈F,(fn)(k)and (gn)(k)share a holomorphic function p(z) in a domain D, then Fis normal in D. And if for (f, g)∈F, L(fn) and L(gn) share a holomorphic function p(z) inD, where L(fn)=b0(fn)(k)+b1(fn)(k1)+...+bkfn, then F is normal in D. |
