On The Deficient Functions, The Singular Directions And Normal Family Theory For Meromorphic Functions

The aim of this work is to consider some problems related to the Deficient theory, the Singular direction theory,the value distribution theory and the normal family theory for meromorphic functions. Our main works are following:1.On Vultrcm Deficient functions of meromorphic functionsFor a function meromorphic on the plane +00 with finite order, we obtain the following result[27]:Theorem 2.1.0.20 Let f(z) be a function rneromorphic on the plane with finite: order, and bo a function meromorphic on theplane, such that and any finite numberof which are independent. Then must be Sct. Where Finally, we also find that the corresponding above conclusion hold for mero-morphic functions with infinite order.Theorem 2.3.0.28 Let f(z) be a function meromorphic on the plane with infinite order, and be a function lneromorphn; withfinite order on the; plane, such that andany finite number of which are independent. SettingThen must be Set.2.On Nevanlinna Directions of Algebroid FunctionsTheorem 3.3.0.31 [25] Suppose that w(z) is v-valued algebriod function defined by equation (2.2.1) in If w(z) satisfiesthen the Nevanlinna direction denned by definition (8.3.0.30) must exist.Theorem 3.7.0.34[25] Let u(z) be an algebriod function in theorem 3.3.0.31. then there is a direction , such thatfor an arbitrary positive uumber and any complex number outside at most 2 + exceptional value a.Theorem 3.7.0.35[25| Suppose that w(z) is v-valued algebriod funtion defined by (3.2.1) in with finite positive order p, , then there is a direction , such thatfor an arbitrary positive nmuber and any complex number outside at most 2 exceptional value a.3. Normality Criteria for Families of Meromorphic FunctionsIn this Chapter, we discuss the value distribution of differential polynomial and consider its corresponding normality criteria .Finally, we also obtain normality for meromorphic functions family sharing multiplicity values and normality of compositions of holomorphic functions and their differential polynomial.(1) Zeros of Differential Polynomial In this part, we mainly study the value distribution of differential polynomial a and discovered some condition under which still takes any nonzero finite complex uumber infinitely times. These results are following[2-1]Theorem 4.1.().36 Let function f(z) be meromorphic and transcendental in the plane, all of whose zeros have multiplicity k+ 2 at least. Then the differential monomial assumes all non-zero finite complex number inlinitely times, where A: is an integer number.Theorem 4.1.0.38 Lot function f(z) be meromorphic and transcendental in the; plane, all of whose zeros have multiplicity k + 1 at least and all of whose pole have multiplicity 2 at leaat. Then the differential monomial f(z)fk(z) of f(z) assumes all non-2010 finite complex number infinitely times, where k is an integer number.(2) Normal Criteria Relation to Differential Polynomial f(z)f(k) (z)-a In reference [24], we get two normality criteria related to differential polynomialTheorem 4.2.1.1 Lot F be a family of meromorphic functions in the domain D, all of whose zeros have multiplicity k -f 2 at least. Suppose that for each then T is a normal family on D, where a is non-zero finite complex number and k 1 is an integer number.Theorem 4.2.1.2 Let F be a family of holomorphic functions in the domain D, all of whose zeros have multiplicity k+ 1 at least. Suppose that for each , then F is a normal family on D, whore a is non-zero finite complex number and k > 1 is an integer number.Theorem 4.2.1.3 Let F bo a family of meromorphic functions in the domain D all of whose zeros have multiplicity k + 1 at least, and of whose polos are multiple. Suppose that for each normal family on D, whore a is non-zero finite complex number and k > 1 is an integer number.(3) Sharing multiplicity values and NormalityIn reference [22], wo study the; normality criteria for families of meromorphic fuuetions sharing multiplicity values with its derivatives and obtain the following conclusionTheorem 4.3.1.1 Let F be a family of meromorphic functi...