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On the other side, Xia and Xu [14] proved the following result:

Theorem A. Let F be a family of meromorphic functions defined on a domain D ⊂ C. Let ψ(6≡ 0) be a in D, and k be a positive integer. Suppose that for every function f ∈ F, f =6 0, f (k) =6 0 and all zeros of f (k) − ψ(z) have multiplicities at k+2 least k . If for k = 1, ψ has only zeros with multiplicities at most 2, and for k ≥ 2, ψ has only simple zeros, then F is normal in D. Recently, B. Deng et al [3] studied the case where f (k) − af n − b has some distinct zeros in D and they proved the following results:

Theorem B. Let F be a family of meromorphic functions in a domain D, let m, n, k be three positive integers such that n ≥ k + m +2, and let a(=6 0), b ∈ C. If for each f ∈F, all zeros of f are of multiplicity at least k and f (k) − af n − b has at most m distinct zeros, then F is normal on D. It is natural to ask if f (k) −af n −ψ has zeros with some multiplicities for a holomorphic function ψ(z) =6 0. In this paper we investigate this situation and prove the following results: Theorem 1.1. Let F be a family of meromorphic functions in a domain D. Let d,k and m k+1 k+2 1 be positive integers such that d + 2m < 2 . Let a(z) and ψ(z)(=6 0) be holomorphic func- tions in D. If for each f ∈F, all zeros of f are of multiplicity at least d, poles of f are multiple and for a positive integer n, all zeros of f (k)(z) − a(z)f n(z) − ψ(z) are of multi- plicity at least m in D, then F is normal in D. For a family of holomorphic functions we have the following strengthened version: Theorem 1.2. Let F be a family of holomorphic functions in a domain D. Let d, k, and m k+1 1 be positive integers such that d + m < 1. Let a(z) and ψ(z)(=6 0) be holomorphic func- tions in D. If for each f ∈F, all zeros of f are of multiplicity at least d and for a positive integer n, all zeros of f (k)(z) − a(z)f n(z) − ψ(z) are of multiplicity at least m in D, then F is normal in D. Corollary 1.3. Let F be a family of holomorphic functions on a domain D, let k, m, n be three positive integers. If for each f ∈ F, f has zeros of multiplicities at least m, all (k) k+1 1 zeros of f − 1 are of multiplicity at least n in D and m + n < 1, then F is normal in D. The following example shows that n can’t be zero in Theorem 1.2. NORMALITY CRITERIA FOR FAMILIES OF MEROMORPHIC FUNCTIONS 3

Example 1.4. Let D := {z ∈ C : |z| < 1}. Consider the family F := {jz5 : j ∈ N} on D. Let a(z) = ez, ψ(z) = −ez. Then for n = 0 and k = 1, f (k)(z) − a(z)f n(z) − ψ(z) has zeros of multiplicity 4. Clearly, all conditions of Theorem 1.2 are satisfied but F is not normal in D. The next example shows that the condition ψ(z) =6 0 can not be relaxed. Example 1.5. Let D := {z ∈ C : |z| < 1}. Consider the family F := {jz4 : j ∈ N} on D. Let a(z) = 0, ψ(z) = zd−1. Then for k = 1, f (k)(z) − a(z)f n(z) − ψ(z) has zeros of multiplicity 3. Clearly, all conditions of Theorem 1.2 are satisfied but F is not normal in D. The following example supports our Theorem 1.2. Example 1.6. Let D := {z ∈ C : z =6 0}. Consider the family F := {jz4 : j ∈ N} on D. Let a(z)= −z−4, ψ(z)= −z2/4. Then for n = 2 and k = 1, f (k)(z)−a(z)f n(z)−ψ(z) has zeros of multiplicity 2 in D and all conditions of Theorem 1.2 are satisfied. F is normal in D. We also investigate this situation for f(z)+a(z)(f (k))n(z)−ψ(z) and prove the following result: Theorem 1.7. Let F be a family of meromorphic functions in a domain D. Let d,k,m nk+1 n(k+1) and n(≥ 2) be four positive integers such that (n−1)d + m(n−1) < 1. Let a(z)(6≡ 0) and ψ(z)(=6 0) be holomorphic functions in D. If for each f ∈ F, all zeros of f are of multiplicity at least d and all zeros of f(z)+ a(z)(f (k))n(z) − ψ(z) are of multiplicity at least m, then F is normal in D. For a family of holomorphic functions we have the following strengthened version: Theorem 1.8. Let F be a family of holomorphic functions in a domain D. Let d,k,m nk+1 1 and n(≥ 2) be four positive integers such that nd + m < 1. Let a(z)(6≡ 0) and ψ(z)(=6 0) be holomorphic functions in D. If for each f ∈ F, all zeros of f are of multiplicity at least d and all zeros of f(z)+ a(z)(f (k))n(z) − ψ(z) are of multiplicity at least m, then F is normal in D. The following example supports Theorem 1.8. Example 1.9. Let D := {z ∈ C : |z| > 0}. Consider the family F := {jz4 : j ∈ N} on D. Let a(z)= z−1/16, ψ(z)= −z3/4. Then for k = 1 and n = 2, f(z)+a(z)(f (k))n(z)−ψ(z) has zeros of multiplicity 2 and all conditions of Theorem 1.8 are satisfied. Clearly, F is normal in D.

2. Some Notation and results of Nevanlinna theory Let ∆ = {z : |z| < 1} be the unit disk. We use the following standard functions of value distribution theory, namely T (r, f), m(r, f), N(r, f) and N(r, f). We let S(r, f) be any function satisfying S(r, f)= o T (r, f) , as r → +∞,  4 S. KUMAR AND P. RANI possibly outside a set of finite measure.

First Fundamental Theorem. Let f be a on C and a be a complex number. Then 1 T r, = T (r, f)+ O(1).  f − a Logarithmic Derivative Lemma. Let f be a non-constant meromorphic function on C, and let k be a positive integer. Then the equality f (k) m r, = S(r, f)  f  holds for all r ∈ [1, ∞) excluding a set of finite measure.

3. Preliminary results In order to prove our results we need the following Lemmas. The well known Zalcman Lemma is a very important tool in the study of normal families. The following is a new version due to Zalcman [20]. Lemma 3.1. [20, 19]Let F be a family of meromorphic functions in the unit disk ∆, with the property that for every function f ∈ F, the zeros of f are of multiplicity at least l and the poles of f are of multiplicity at least k . If F is not normal at z0 in ∆, then for −l<α

(1) a sequence of complex numbers zn → z0, |zn|

αi(z) 6≡ 0 are small functions of f, that is T (r, αi)= o T (r, f) . The lower degree of the differential polynomial P is defined by  p

d(P ) := min Sij. 1≤i≤n Xj=0 The following result was proved by Dethloff et al. in [4].

Lemma 3.4. Let a1,...,aq be distinct non-zero complex numbers. Let f be a non- constant meromorphic function and let P be a non-constant differential polynomial of f with d(P ) ≥ 2. Then q qθ(P )+1 1 1 1 T (r, f) ≤ N r, + N r, + S (r, f) , qd(P ) − 1  f  qd(P ) − 1  P − aj  Xj=1 p for all r ∈ [1, +∞) excluding a set of finite Lebesgue measure, where θ(P ) := max jSij. 1≤i≤n Xj=0 Moreover, in the case of an entire function, we have q qθ(P )+1 1 1 1 T (r, f) ≤ N r, + N r, + S(r, f),  qd(P )   f  qd(P )  P − aj  Xj=1 for all r ∈ [1, +∞) excluding a set of finite Lebesgue measure. This result was proved by Hinchliffe in [10] for q = 1.

4. Proof of Main Results Proof of Theorem 1.1: Since normality is a local property, we assume that D = ∆. Suppose that F is not normal in ∆. Then there exists at least one point z0 such that F is not normal at the point z0 in ∆. Without loss of generality we assume that z0 = 0. Then by Lemma 3.1, for α = −k there exist

(1) a sequence of complex numbers zj → 0, |zj|

We see that (k) n fj (zj + ρjζ) − a(zj + ρjζ)fj (zj + ρjζ) − ψ(zj + ρjζ) (k) nk = gj (ζ) − ρj a(zj + ρjζ)gj(ζ) − ψ(zj + ρjζ) → g(k)(ζ) − ψ(0). 6 S. KUMAR AND P. RANI

By Hurwitz’s Theorem, we see that g(k)(ζ) − ψ(0) has at least m zeros. Now, by Milloux’s inequality and Nevanlinna’s First Fundamental Theorem we get 1 1 1 T (r, g) ≤ N(r, g)+ N r, + N r, − N r, + S(r, g)  g   g(k) − ψ(0)  g(k+1)  1 1 1 ≤ N(r, g)+(k + 1)N r, + N r, + S(r, f) 2  g   g(k) − ψ(0) 1 k +1 1 1 1 ≤ N(r, g)+ N r, + N r, + S(r, g) 2 d  g  m  g(k) − ψ(0) 1 k +1 1 ≤ + T (r, g)+ T (r, g)+ kN(r, g) + S(r, g) 2 d  m  1 k +1 1 k ≤ + + + T (r, g)+ S(r, g). 2 d m 2m k+1 k+2 1 Combining with assumption d + 2m < 2 , we get g is constant. This is a contradic- tion. Hence F is a normal family, and this completes the proof of Theorem 1.1. 

We can prove Theorem 1.2 by the method of Theorem 1.1. Since F is a family of holomorphic functions, so by Remark 3.2, the limit function g is a non-constant entire function with zeros of multiplicity at least d and g(k)(ζ) − ψ(0) has at least m zeros. Now, by Lemma 3.3 and Nevanlinna’s First Fundamental Theorem we get 1 1 1 T (r, g) ≤ N(r, g)+ N r, + N r, − N r, + S(r, g)  g   g(k) − ψ(0)  g(k+1)  1 1 ≤ (k + 1)N r, + N r, + S(r, f)  g   g(k) − ψ(0) k +1 1 1 1 ≤ N r, + N r, + S(r, g) d  g  m  g(k) − ψ(0) k +1 1 ≤ T (r, g)+ T (r, g)+ kN(r, g) + S(r, g) d m  k +1 1 ≤ + T (r, g)+ S(r, g).  d m k+1 1 Combining with assumption d + m < 1, we get g is constant. This is a contradiction. Hence F is a normal family, and this completes the proof of the Theorem 1.2. 

Proof of Theorem 1.7: Again we assume that D = ∆. Suppose that F is not normal in ∆. Then there exists at least one point z0 such that F is not normal at the point z0 in ∆. Without loss of generality we assume that z0 = 0. Then by Lemma 3.1, for α = −k there exist

(1) a sequence of complex numbers zj → 0, |zj|

We see that n (k) fj(zj + ρjζ)+ a(zj + ρjζ) fj (zj + ρjζ) − ψ(zj + ρjζ)   n k (k) = ρj gj(ζ)+ a(zj + ρjζ) gj (ζ) − ψ(zj + ρjζ) n   → a(0) g(k) (ζ) − ψ(0).

 n Let a(0) = a. By Hurwitz’s Theorem, we see that a g(k) (ζ) − ψ(0) has at least m n zeros. Now we invoke Lemma 3.4 for the differential polynomial  P = a g(k) (ζ) and q = 1. Note that, d(P )= n(≥ 2) and θ(P )= nk. By Lemma 3.4 and Nevanlinna’s  First Fundamental Theorem, nk +1 1 1 1 T (r, g) ≤ N r, + N r, + S(r, g) n − 1  g  n − 1  a (g(k))n − ψ(0) nk +1 1 1 1 ≤ N r, + N r, + S(r, g) (n − 1)d  g  (n − 1)m  a (g(k))n − ψ(0) nk +1 1 1 n ≤ N r, + T r, g(k) + S(r, g) (n − 1)d  g  (n − 1)m    nk +1 1 n = N r, + T r, g(k) + S(r, g) (n − 1)d  g  (n − 1)m  nk +1 1 n = N r, + m r, g(k) + N r, g(k) + S(r, g) (n − 1)d  g  (n − 1)m   nk +1 n g(k) ≤ T (r, g)+ m r, + m (r, g) (n − 1)d (n − 1)m   g   n + N(r, g)+ kN(r, g) + S(r, g) (n − 1)m  nk +1 n nk ≤ T (r, g)+ T (r, g)+ T (r, g)+ S(r, g) (n − 1)d (n − 1)m (n − 1)m nk +1 n(1 + k) ≤ + T (r, g)+ S(r, g). (n − 1)d (n − 1)m nk +1 n(1 + k) Combining this with assumption + < 1, we get that g is a constant. (n − 1)d (n − 1)m This is a contradiction. Hence F is a normal family and this completes the proof of The- orem 1.7. 

Again, we can prove Theorem 1.8 by the method of Theorem 1.7. Since F is a family of holomorphic functions so by Remark 3.2, the limit function g is a non-constant entire n function with zeros of multiplicity at least d and a g(k) (ζ) − ψ(0) has at least m zeros. n Now, we apply Lemma 3.4 to the differential polynomial P = a g(k) (ζ) and q = 1. Note that, d(P )= n(≥ 2) and θ(P )= nk. By Lemma 3.4 and Nevanlinna’s  First Fundamental 8 S. KUMAR AND P. RANI

Theorem, nk +1 1 1 1 T (r, g) ≤ N r, + N r, + S(r, g) n  g  n  a (g(k))n − ψ(0) nk +1 1 1 1 ≤ N r, + N r, + S(r, g) nd  g  nm  a (g(k))n − ψ(0) nk +1 1 1 n ≤ N r, + T r, g(k) + S(r, g) nd  g  nm    nk +1 1 n = N r, + T r, g(k) + S(r, g) nd  g  nm  nk +1 1 1 = N r, + m r, g(k) + N r, g(k) + S(r, g) nd  g  m   nk +1 1 g(k) ≤ T (r, g)+ m r, + m (r, g) + S(r, g) nd m   g   nk +1 1 ≤ T (r, g)+ T (r, g)+ S(r, g) nd m nk +1 1 ≤ + T (r, g)+ S(r, g).  nd m nk +1 1 Combining this with assumption + < 1, we get g is constant. This is a con- nd m tradiction. Hence F is a normal family. This completes the proof of the Theorem 1.8. 

Acknowledgement: The second author is thankful to the faculty and the adminis- trative unit of School of Mathematics, Harish-Chandra Research Institute, Allahabad for their warm hospitality during the preparation of this paper.

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Department of Mathematics, Deen Dayal Upadhyaya College, University of Delhi, Delhi–110 078, India E-mail address: [email protected] Department of Mathematics, University of Delhi, Delhi–110 007, India E-mail address: [email protected]