An Analysis on Fix Points Common and Natural Families of Some Homogeneous Polynomials

The study of Fix points plays an important role in the criteria of Normality of meromorphic functions. The connection between Fix-points and Normality criteria was given by the following theorem of Yong Lo. Theorem: Let F be a meromorphic function family in area D, and let k be a positive integer. If for some f(z) of F function, Both f and f and Don't have fix points in D, Then there is regular F. C.M. Hombale extended this result to certain homogeneous differential polynomials and proved the following theorem Theorem : Let F be a family of meromorphic functions in a region D, k be positive integer if i) For every , f has only multiple poles and at every double pole z0, the Laurent expansion of f(z) has the form ii) For every , and (the derivative of order k) Don't have fix points in D, Then there is regular F. Here, In that segment, We don't just expand the theorem B above, but, also remove the condition (i), we adopt a different technique, and here we prove, Theorem: Let F be a non-zero meromorphic family of functions in a region D, , K are positive results, if any (the derivative of order f(z)) Don't have repair points in D, then F's natural. Let Lemma: Let f(z) have meromorphic in . If after that, They've got Proof: Consider the identity This leads to Now since Hence Similarly And Hence we obtain Adding on both sides, and by using first fundamental theorem of Nevanlinna, we note that Thus, we obtain Now Thus, by combining, (4) and (3), we get Where Lemma: Suppose, f(z) is as given in lemma, and Then

But, since We have Thus Substituting this inequality in (5) we get Thus Lemma: Suppose f(z) satisfies the assumptions of Lemma, with and in addition, then, we have for where C is a positive numerical constant and Proof: By hypothesis, we note that

We have proved a result on The singularity of meromorphic functions. But we do prove in this part a result concerning normality criterion for a family of meromorphic functions. We have Theorem: Let F be a family of meromorphic functions having only multiple poles in D, such that for each in D for any fixed integer . In D, then, F is normal. We need the following lemmas for proof Lemma: (Heiong estimate) Suppose that f(z) is meromorphic in and that. Then, For Holds for every pair of then we have Lemma: Suppose is a sequence of meromorphic functions on and for all . if for and . then there exists a neighborhood of 0 on which some subsequence of tends to an analytic function.

Let f (z) is a meromorphic function in a domain D. We assume familiarity with usual notations of Nevanlinna theory. Throughout this we use to denote the fixed constants depending at most on a, We also let where are non-negative integers, we call as the degree of the monomial M(z) and as it's weight. In the theory of normal families, A major problem is finding new standards of normality. Nevanlinna theory plays a significant role in that regard. The following theorem was proved by Langley. J.K using the Nevanlinna theory. Let F be a meromorphic family of functions in domain D and for fixed functions , and for some , then F is normal in D. We prove a significant generalization of Langley's result. For the proof, we use the methods of both Langley.J.K, Xu.Y and X.Hua, This approach greatly simplifies the proofs. They show the following principal theorem. Theorem: Let F become a meromorphic family of functions within a domain. Suppose a, b are complex and finite numbers with and for fixed integer , for each , , where with F in D then is natural. Lemmas Lemma: Let F be a meromorphic function family within a Domain D, such that for each has no solutions there ( where a, b are finite, and ) Suppose further that and ; where Then for Proof: - Since Hence, Therefore, Noting that We have, Using the standard estimate (5) for ' we have Also, If we denote by the counting function of the common zeros of both K and M(z) - b, we rewrite the above equation as, Where in , we count the distinct common zeros of K and M(z) - b. Obviously, A zero of K is either a pole of f or a zero of M(z)-b and a pole of f of order p must be a zero of K of order p . Hence, So,

fundamental theorem As. Also it is easy to see that, From the above discussion and by equation (3), We have, This gives, Now using the standard estimate [5] for We can write Now since, we have, Therefore, Substituting this in equation (5), we get, Where Now applying Bureau's lemma in the interval (0,S) we obtain, We need lemma due to Ku Lemma: Suppose that f (z) is meromorphic in with . Suppose that for . Then f is regular in and in. Proof of Theorem: Given a point , we take a positive R such that . Without loss of generality we may assume that . We set where D and E are constants of by Lemma. Now by Lemma, we conclude that,| And so by lemma, f (z) is regular in and in . We settle for a small positive so that . And so in . Hence by Montel's theorem, we conclude that F is normal in D. Further we obtain the following result of Langley as a special case of theorem. Corollary: Let F be a family of meromorphic functions in a domain D, for fixed are finite complex numbers with and has no solution in D, then F is normal in D. Proof of Corollary: Here , we note that . Therefore, taking we arrive at the result of Langley.

In [8] Hayman‘s problem is related to the where and . Hayman [8] Demonstrated meromorphic function on C that satisfies Should be continuous, if . If f is entire then the result is' true for . For analytic functions, the normality result corresponding to s Hayman‘s theorem was proved by Drasin D[3], The corresponding result for meromorphic functions was established (independently) by Langley.J.K, Xianjin Li etc. Now, we prove the following theorem using Zalcman lemma. Here, we take, where are small holomorphic functions (unless otherwise stated) and each is a monomial generated by f. As usual and denote degree and weight of respectively. We also take, as the term with highest degree and weight among and hence and . in the complex plane which has only poles of order at least p, a be a non-zero finite complex number with and also . Suppose that f is not a polynomial of degree less than k, then Where, as possibly outside a set of finite linear measure. Proof: It is easy to see that

Let f be a meromorphic function. Let us define a monomial in f, by where are all positive integers. We call the degree and the weight of the monomial . Let where are constants and for convince we write Then Is called polynomial differential with degree in f and weight . condition and for then Is called the homogeneous polynomial derivative in f. The concerns Hayman has about normal families are all of similar form. In each case, it is understood that a property concerning the values of a function and its derivatives means that a whole or globally defined meromorphic function has to be constant. Does a family of meromorphic functions have the same property imply normality? Hayman proved that an entire or a meromorphic function which satisfies for fixed must be a constant. The corresponding results on normal families were proved by Yang Lo. Recently Ming-Liang Fang and W. Hong proved the following theorem:

functions in a domain D be two positive integers, be a differential polynomial in f and . If the zeros of f(z) are of multiplicity and for each then F is normal in D. Now we prove the following two theorems. Theorem: Let F be a family of meromorphic functions in a domain D such that each satisfies for any then F is normal in D, where is a homogeneous differential polynomial and . For the proof we need the following important lemmas. As an application of Theorem we deduce the following Lemma Lemma: Suppose f(z) is a transcendental meromorphic function in Is a monomial in f ' and not identically constant then f(z) infinitely always assumes any finite value or Assumes infinitely always every finite, non-zero value. Proof of Theorem: Denote F to the family in question, and assume F is not natural. One may presume as normal select By Lemma, It exists A list of positive numbers So that converges to locally - uniformly where g is a non-constant meromorphic function in D. Since and g is a non-constant by Hurwitz‘s theorem. Also is the uniform limit of Where Thus and converges to imply, either or Case: If then for all which is impossible. Case: If then we arrive at a contradiction to Lemma. Consequently F is natural in D. Theorem: Let F be a meromorphic functional family having zeros of order at least k, in a domain in D. Suppose that there exists a constant such that, for each . Whenever then F is normal in D. Proof: Let and take obviously When F isn't normal at Instead, there is a series of Lemma Positive figures and a sequence such that converges to a non-constant meromorphic function spherically and locally uniformly in D. Also since g is not a polynomial of degree k, if g is a rational function or transcendental function then also has zeros. Thus there exists a, such that Then for large m and there exists a positive constant such that On the other hand, there exists a natural number such that for But it is the uniform limit of Hence, we have, Thus, by Rouche‘s theorem, there exists a point such that Combining this with the hypothesis, we have Without loss of generality, we suppose that then Now the left hand side of (4) converges to and the right side of (4) tends to by the fact that we have . Let then Thus which contradicts (3). Hence the proof.

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PhD Student, MUIT, Lucknow