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Some results on the value distribution of differential polynomials

  • Jinyu Fan , Jianbin Xiao and Mingliang Fang EMAIL logo
Published/Copyright: March 14, 2023
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Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

In this article, we study some results on the value distribution of differential polynomials and mainly prove the following theorem: suppose that P is a polynomial with deg P ≥ 3 and f is a transcendental meromorphic function. Let α be a small function of f . If α is a constant, we also require that there exists a constant A ≠ α such that P ( z ) − A has a zero of multiplicity at least 3. Then, for any 0 < ε < 1 , we have

T ( r , f ) ≤ k N ¯ r , 1 P ( f ) − α + S ( r , f ) ,

where if P ′ ( z ) has only one zero, then k = 1 deg P − 2 ; if P ′ ( z ) has two distinct zeros a and b with P ( a ) ≠ P ( b ) and α is nonconstant, then k = 1 1 − ε ; otherwise k = 1 .

Keywords: meromorphic function; small function; deficiency value
MSC 2010: 30D35

1 Introduction and main results

In this article, a meromorphic function always means meromorphic in the whole complex plane. We use the following standard notations in value distribution theory, see [1–3]: T ( r , f ) , N ( r , f ) , m ( r , f ) , … .

We denote by S ( r , f ) any quantity satisfying S ( r , f ) = o ( T ( r , f ) ) as r → ∞ , which is outside of an exceptional set E with finite measure. A meromorphic function α is said to be a small function of f if it satisfies T ( r , α ) = S ( r , f ) .

Let f be a nonconstant meromorphic function. Define

Θ ( ∞ , f ) = 1 − lim ¯ r → ∞ N ¯ ( r , f ) T ( r , f )

and

δ ( ∞ , f ) = 1 − lim ¯ r → ∞ N ( r , f ) T ( r , f ) .

Let f be a nonconstant meromorphic function, let j be a positive integer, let n 0 j , n 1 j , … , n k j be non-negative integers, and let a j ( z ) be a small function of f . We call that M j ( f ) = a j ( z ) ( f ( z ) ) n 0 j ( f ′ ( z ) ) n 1 j ⋯ ( f ( k ) ( z ) ) n k j is a differential monomial of f . We define d ( M j ) = ∑ i = 0 k n i j as the degree of M j ( f ) . Next, a differential polynomial in f is a finite sum of such monomials, i.e.,

(1) φ ( f ) = ∑ n j = 0 M j ( f ) .

We define d ( φ ) = max 0 ≤ j ≤ n { d ( M j ) } and d ̲ ( φ ) = min 0 ≤ j ≤ n { d ( M j ) } as the degree and lower degree of φ ( f ) , respectively.

In 1952, Rosenbloom [4] proved the following theorem.

Theorem A

Let P be a polynomial with deg P ≥ 2 , and let f be a transcendental entire function. Then,

lim ¯ r → ∞ N ¯ r , 1 P ( f ) − z T ( r , f ) ≥ 1 .

From Theorem A, we obtain

Theorem B

Let P be a polynomial with deg P ≥ 2 , and let f be a transcendental entire function. Then, P ( f ) has infinitely many fix-points.

In 1972, Gross and Yang [5] extended Theorem B to the case of meromorphic functions.

Theorem C

Let P be a polynomial with deg P ≥ 3 , and let f be a transcendental meromorphic function. Then, P ( f ) has infinitely many fix-points.

In 1995, Zheng and Yang [6] proved the following theorem.

Theorem D

Suppose that P is a polynomial with deg P ≥ 2 and f is a transcendental entire function. Let α be a nonconstant small function of f. Then,

T ( r , f ) ≤ k N ¯ r , 1 P ( f ) − α + S ( r , f ) ,

where k = 2 deg P − 1 if P ′ ( z ) has only one zero; otherwise k = 2 .

In 2000, Fang and Yuan [7] improved the above results and proved the following theorem.

Theorem E

Suppose that P is a polynomial with deg P ≥ 2 and f is a transcendental entire function. Let α be a small function of f. If α is a constant, we also require that there exists a constant A ≠ α such that P ( z ) − A has a zero of multiplicity at least 2. Then,

T ( r , f ) ≤ k N ¯ r , 1 P ( f ) − α + S ( r , f ) ,

where k = 1 deg P − 1 if P ′ ( z ) has only one zero; otherwise k = 1 .

Naturally, we pose the following problem.

Problem 1

Whether the entire functions of Theorem E can be replaced by meromorphic functions?

In this article, we study this problem and prove the following result.

Theorem 1

Suppose that P is a polynomial with deg P ≥ 3 and f is a transcendental meromorphic function. Let α be a small function of f . If α is a constant, we also require that there exists a constant A ≠ α such that P ( z ) − A has a zero of multiplicity at least 3. Then, for any 0 < ε < 1 , we have

T ( r , f ) ≤ k N ¯ r , 1 P ( f ) − α + S ( r , f ) ,

where if P ′ ( z ) has only one zero, then k = 1 deg P − 2 ; if P ′ ( z ) has two distinct zeros a and b with P ( a ) ≠ P ( b ) , and α is nonconstant, then k = 1 1 − ε ; otherwise k = 1 .

The following example shows that the condition deg P ≥ 3 in Theorem 1 is sharp.

Example 1

Let P ( z ) = z 2 , f − z f + z = e z and α ( z ) = z 2 , then we have f ( z ) = z e z + z 1 − e z and

P ( f ) − α = ( f − z ) ( f + z ) ≠ 0 .

Hence, we obtain

N ¯ r , 1 P ( f ) − α = 0 .

The following example shows that the condition A ≠ α in Theorem 1 is necessary.

Example 2

Let P ( z ) = ( z − 1 ) 3 ( z − 2 ) + 1 , f ( z ) = 2 e z − 1 e z − 1 , and α = A = P ( 1 ) = 1 . Hence, we obtain

N ¯ r , 1 P ( f ) − α = 0 .

The following example shows that it is necessary that P ( z ) − A has a zero of multiplicity at least 3 in Theorem 1.

Example 3

Let P ( z ) = z 3 − 6 z 2 + 9 z + 1 , f ( z ) = 4 e z − 1 e z − 1 , α = P ( 1 ) = 5 , and A = P ( 3 ) = 1 . Hence, we obtain

N ¯ r , 1 P ( f ) − α = 0 .

In 1959, Hayman [8] proved the following result.

Theorem F

Let f be a transcendental entire function and n ( ≥ 2 ) be a positive integer. Then, f n ( z ) f ′ ( z ) assumes all finite nonzero values infinitely often.

In 1967, Clunie [9] proved that Theorem F was true for n = 1 .

In 1969, Sons [10] generalized Theorem F and obtained the following result.

Theorem G

Let f be a transcendental entire function, and let

(2) φ ( f ( z ) ) = ( f ( z ) ) n 0 ( f ′ ( z ) ) n 1 ⋯ ( f ( k ) ( z ) ) n k .

  1. If φ has the form (2), where n 0 ≥ 2 , n k ≥ 1 , and n i ≥ 0 for i ≠ 0 , k , then δ ( a , φ ) < 1 for all a ≠ 0 , ∞ .

  2. If f satisfies N 1 ( r , 1 f ) = o { T ( r , f ) } , where N 1 ( r , 1 f ) is the counting function of simple zeros of f , and φ has the form (2), where n 0 ≥ 1 , n k ≥ 1 , and n i ≥ 0 for i ≠ 0 , k , then δ ( a , φ ) < 1 for all a ≠ 0 , ∞ .

Yang proved the following theorems.

Theorem H

[11] Suppose that f is a transcendental meromorphic function with

N ( r , f ) + N r , 1 f = S ( r , f ) .

Let φ ( f ) be a differential polynomial in f of the form (1) with no constant term. Furthermore, we assume that the degree n ( ≥ 2 ) of φ ( f ) and l 0 < n , 0 ≤ l i ≤ n for all i ≠ 0 . Then, δ ( a , φ ( f ) ) < 1 for all a ≠ 0 , ∞ .

Theorem I

[12] Suppose that f is a transcendental meromorphic function with

N ( r , f ) + N r , 1 f = S ( r , f ) .

Let φ ( f ) be a differential polynomial in f of the form (1) of degree n ≥ 2 such that all terms in φ ( f ) have a degree at least two. Then, if φ ( f ) consists of terms of different degrees, i.e., φ ( f ) is not homogeneous, we have

δ ( a , φ ( f ) ) ≤ 1 − 1 2 n

for all a ≠ ∞ .

In 2008, Bhoosnurmath et al. [13] proved the following result.

Theorem J

Suppose that f is a transcendental meromorphic function with

N ( r , f ) + N r , 1 f = S ( r , f ) .

Let φ ( f ) be a differential polynomial in f of the form (1) of degree d ( φ ) and the lower degree d ̲ ( φ ) . Assume that φ ( f ) does not reduce to a constant.

  1. If φ ( f ) is a homogeneous differential polynomial, then we have

    δ ( a , φ ( f ) ) = 0

    for any a ≠ 0 , i.e., φ ( f ) assumes all finite nonzero complex values infinitely often.

  2. If φ ( f ) is a non-homogeneous differential polynomial with 2 d ̲ ( φ ) > d ( φ ) , then we have

    δ ( a , φ ( f ) ) ≤ 1 − 2 d ̲ ( φ ) − d ( φ ) d ̲ ( φ ) < 1

    for any a ≠ 0 , i.e., φ ( f ) assumes all finite nonzero complex values infinitely often.

Naturally, we pose the following problem.

Problem 2

Whether the results of Theorems H–J can be improved or not?

In this article, we study Problem 2 and prove the following theorem.

Theorem 2

Suppose that f is a transcendental meromorphic function with

(3) N ( r , f ) + N r , 1 f = S ( r , f ) .

Let φ ( f ) be a differential polynomial in f of the form (1) of degree d ( φ ) ≥ 2 and the lower degree d ̲ ( φ ) ≥ 1 . Assume that φ ( f ) does not reduce to a constant.

  1. If φ ( f ) is a homogeneous differential polynomial, then we have

    Θ ( a , φ ( f ) ) = 0 ,

    where a ( ≢ 0 ) is a small function of f .

  2. If φ ( f ) is a non-homogeneous differential polynomial, then we have

    (4) Θ ( a , φ ( f ) ) ≤ 1 − d ̲ ( φ ) d ( φ ) ,

    where a ( ≢ 0 ) is a small function of f .

The following example shows that the bound 1 − d ̲ ( φ ) d ( φ ) in (4) is sharp.

Example 4

Let f = e z , φ ( f ) = f 4 − 2 f f ′ , and a = − 1 , then we have

(5) N ¯ r , 1 φ ( f ) − a = N ¯ r , 1 e 2 z − 1 ≤ 2 T ( r , e z ) + S ( r , f ) .

By Lemma 1, we obtain

(6) 2 T ( r , e z ) = T ( r , e 2 z ) ≤ N ¯ ( r , e 2 z ) + N ¯ r , 1 e 2 z + N ¯ r , 1 e 2 z − 1 + S ( r , f ) ≤ N ¯ r , 1 e 2 z − 1 + S ( r , f ) .

It follows from (5) and (6) that

(7) N ¯ r , 1 φ ( f ) − a = 2 T ( r , e z ) + S ( r , f ) .

Hence, by Lemma 3 and (7), we obtain

Θ ( a , φ ( f ) ) = 1 − d ̲ ( φ ) d ( φ ) = 1 2 .

2 Lemmas

For the proof of our results, we need the following lemmas.

Lemma 1

[1–3] Let f be a nonconstant meromorphic function, and let a i ( i = 1 , 2 ) be two distinct small functions of f. Then,

T ( r , f ) ≤ N ¯ ( r , f ) + N ¯ r , 1 f − a 1 + N ¯ r , 1 f − a 2 + S ( r , f ) .

Lemma 2

[14] Let f be a nonconstant meromorphic function, and let a i ( i = 1 , 2 , 3 ) be three distinct small functions of f. Then, for any 0 < ε < 1 , we have

2 T ( r , f ) ≤ N ¯ ( r , f ) + ∑ 3 i = 1 N ¯ r , 1 f − a i + ε T ( r , f ) + S ( r , f ) .

Lemma 3

[12] Let P ( z ) = a n z n + a n − 1 z n − 1 + ⋯ + a 1 z + a 0 , where a n ( ≠ 0 ) , a n − 1 , … , a 0 are constants. If f is a nonconstant meromorphic function, then

T ( r , P ( f ) ) = n T ( r , f ) + S ( r , f ) .

Lemma 4

[1–3] Let f be a nonconstant meromorphic function, and let k be a positive integer. Then,

m r , f ( k ) f = S ( r , f ) .

3 Proof of Theorem 1

Set n = deg P ≥ 3 . We consider two cases.

Case 1. α is a nonconstant small function of f . Next, we consider two subcases.

Case 1.1. P ′ ( z ) has only a zero a . Set A = P ( a ) , then we have

(8) P ( z ) − A = c 1 ( z − a ) n ,

where c 1 is a nonzero constant.

By Lemma 1 and (8), we obtain

(9) T ( r , P ( f ) ) ≤ N ¯ ( r , P ( f ) ) + N ¯ r , 1 P ( f ) − A + N ¯ r , 1 P ( f ) − α + S ( r , P ( f ) ) ≤ N ¯ ( r , f ) + N ¯ r , 1 f − a + N ¯ r , 1 P ( f ) − α + S ( r , f ) ≤ 2 T ( r , f ) + N ¯ r , 1 P ( f ) − α + S ( r , f ) .

It follows from Lemma 3 and (9) that

( n − 2 ) T ( r , f ) ≤ N ¯ r , 1 P ( f ) − α + S ( r , f ) .

That is,

T ( r , f ) ≤ 1 deg P − 2 N ¯ r , 1 P ( f ) − α + S ( r , f ) .

Case 1.2. P ′ ( z ) has at least two zeros a , b ( a ≠ b ) , where the multiplicity of a , b is m 1 , m 2 , respectively, and m 1 , m 2 ≥ 1 . Set A = P ( a ) and B = P ( b ) . In the following, we consider two subcases.

Case 1.2.1. A ≠ B . We have

(10) P ( z ) − A = c 2 ( z − a ) m 1 + 1 ϕ 1 ( z ) , P ( z ) − B = c 3 ( z − b ) m 2 + 1 ϕ 2 ( z ) ,

where c 2 and c 3 are two nonzero constants and ϕ 1 ( z ) , ϕ 2 ( z ) are two polynomials with deg ϕ 1 = n − m 1 − 1 and deg ϕ 2 = n − m 2 − 1 .

By Lemma 2 and (10), we obtain

(11) 2 T ( r , P ( f ) ) ≤ N ¯ ( r , P ( f ) ) + N ¯ r , 1 P ( f ) − A + N ¯ r , 1 P ( f ) − B + N ¯ r , 1 P ( f ) − α + ε n T ( r , P ( f ) ) + S ( r , P ( f ) ) ≤ N ¯ ( r , f ) + N ¯ r , 1 f − a + ( n − m 1 − 1 ) T ( r , f ) + N ¯ r , 1 f − b + ( n − m 2 − 1 ) T ( r , f ) + N ¯ r , 1 P ( f ) − α + ε T ( r , f ) + S ( r , f ) ≤ ( 2 n − m 1 − m 2 + 1 + ε ) T ( r , f ) + N ¯ r , 1 P ( f ) − α + S ( r , f ) ≤ ( 2 n − 1 + ε ) T ( r , f ) + N ¯ r , 1 P ( f ) − α + S ( r , f ) .

It follows from Lemma 3 and (11) that

( 1 − ε ) T ( r , f ) ≤ N ¯ r , 1 P ( f ) − α + S ( r , f ) .

That is,

T ( r , f ) ≤ 1 1 − ε N ¯ r , 1 P ( f ) − α + S ( r , f ) .

Case 1.2.2. A = B . We have

(12) P ( z ) − A = c 4 ( z − a ) m 1 + 1 ( z − b ) m 2 + 1 ϕ 3 ( z ) ,

where c 4 is a nonzero constant and ϕ 3 ( z ) is a polynomial with deg ϕ 3 = n − m 1 − m 2 − 2 .

By Lemma 1 and (12), we obtain

(13) T ( r , P ( f ) ) ≤ N ¯ ( r , P ( f ) ) + N ¯ r , 1 P ( f ) − A + N ¯ r , 1 P ( f ) − α + S ( r , P ( f ) ) ≤ N ¯ ( r , f ) + N ¯ r , 1 f − a + N ¯ r , 1 f − b + ( n − m 1 − m 2 − 2 ) T ( r , f ) + N ¯ r , 1 P ( f ) − α + S ( r , f ) ≤ ( n − m 1 − m 2 + 1 ) T ( r , f ) + N ¯ r , 1 P ( f ) − α + S ( r , f ) ≤ ( n − 1 ) T ( r , f ) + N ¯ r , 1 P ( f ) − α + S ( r , f ) .

It follows from Lemma 3 and (13) that

T ( r , f ) ≤ N ¯ r , 1 P ( f ) − α + S ( r , f ) .

Case 2. α is a constant function. Then there exists a constant A ≠ α such that P ( z ) − A has a zero of multiplicity at least 3. Hence, we have

(14) P ( z ) − A = c 5 ( z − a ) m ϕ 4 ( z ) ,

where m ≥ 3 , c 5 is a nonzero constant and ϕ 4 ( z ) is a polynomial with deg ϕ 4 = n − m .

By Lemma 1 and (14), we obtain

(15) T ( r , P ( f ) ) ≤ N ¯ ( r , P ( f ) ) + N ¯ r , 1 P ( f ) − A + N ¯ r , 1 P ( f ) − α + S ( r , P ( f ) ) ≤ N ¯ ( r , f ) + N ¯ r , 1 f − a + ( n − m ) T ( r , f ) + N ¯ r , 1 P ( f ) − α + S ( r , f ) ≤ ( n − m + 2 ) T ( r , f ) + N ¯ r , 1 P ( f ) − α + S ( r , f ) ≤ ( n − 1 ) T ( r , f ) + N ¯ r , 1 P ( f ) − α + S ( r , f ) .

It follows from Lemma 3 and (15) that

T ( r , f ) ≤ N ¯ r , 1 P ( f ) − α + S ( r , f ) .

This completes the proof of Theorem 1.

3.1 Proof of Theorem 2

From the assumption and (1), we know that φ ( f ) ≢ 0 .

In the following, we consider two cases.

Case 1. φ ( f ) is a homogeneous differential polynomial. Then, by (1), we have

(16) φ ( f ) = ∑ n j = 0 a j ( z ) ( f ( z ) ) n 0 j ( f ′ ( z ) ) n 1 j ⋯ ( f ( k ) ( z ) ) n k j = ∑ n j = 0 a j ( z ) ( f ( z ) ) n 0 j + n 1 j + ⋯ + n k j f ′ ( z ) f ( z ) n 1 j ⋯ f ( k ) ( z ) f ( z ) n k j = ∑ n j = 0 b j ( z ) ( f ( z ) ) d ( φ ) = B ( z ) ( f ( z ) ) d ( φ ) ,

where b j ( z ) = a j ( z ) f ′ ( z ) f ( z ) n 1 j ⋯ f ( k ) ( z ) f ( z ) n k j and B ( z ) = ∑ j = 0 n b j ( z ) .

Since a j ( z ) is a small function of f , by Lemma 4 and (3), we obtain

m ( r , b j ) = m r , a j f ′ f n 1 j ⋯ f ( k ) f n k j ≤ m ( r , a j ) + m r , f ′ f n 1 j ⋯ f ( k ) f n k j = S ( r , f ) ,

and

N ( r , b j ) = N r , a j f ′ f n 1 j ⋯ f ( k ) f n k j ≤ N ( r , a j ) + N r , f ′ f n 1 j ⋯ f ( k ) f n k j ≤ O N ¯ ( r , f ) + N ¯ r , 1 f + S ( r , f ) ≤ O N ( r , f ) + N r , 1 f + S ( r , f ) = S ( r , f ) .

Hence, we obtain

(17) T ( r , b j ) = m ( r , b j ) + N ( r , b j ) = S ( r , f ) , T ( r , B ) = T r , ∑ n j = 0 b j = S ( r , f ) .

By (3), (16), (17), and Lemma 1, we have

(18) T ( r , φ ( f ) ) ≤ N ¯ ( r , φ ( f ) ) + N ¯ r , 1 φ ( f ) + N ¯ r , 1 φ ( f ) − a + S ( r , φ ( f ) ) ≤ N ¯ ( r , B f d ( φ ) ) + N ¯ r , 1 B f d ( φ ) + N ¯ r , 1 φ ( f ) − a + S ( r , φ ( f ) ) ≤ N ¯ r , 1 φ ( f ) − a + S ( r , φ ( f ) ) .

It follows from (18) that

1 ≤ lim ¯ r → ∞ N ¯ r , 1 φ ( f ) − a T ( r , φ ( f ) ) + lim ̲ r → ∞ S ( r , φ ( f ) ) T ( r , φ ( f ) ) = lim ¯ r → ∞ N ¯ r , 1 φ ( f ) − a T ( r , φ ( f ) ) .

Hence, we obtain Θ ( a , φ ( f ) ) = 0 .

Case 2. φ ( f ) is a non-homogeneous differential polynomial. Then by (1), we have

(19) φ ( f ) = ∑ n j = 0 a j ( z ) ( f ( z ) ) n 0 j ( f ′ ( z ) ) n 1 j ⋯ ( f ( k ) ( z ) ) n k j = ∑ n j = 0 a j ( z ) ( f ( z ) ) n 0 j + n 1 j + ⋯ + n k j f ′ ( z ) f ( z ) n 1 j ⋯ f ( k ) ( z ) f ( z ) n k j = ∑ n j = 0 b j ( z ) ( f ( z ) ) l j = ∑ s i = 0 d i ( z ) ( f ( z ) ) m i ,

where s ≤ n , b j ( z ) = a j ( z ) f ′ ( z ) f ( z ) n 1 j ⋯ f ( k ) ( z ) f ( z ) n k j , l j = n 0 j + n 1 j + ⋯ + n k j , d i ( z ) ≢ 0 , and m 0 , m 1 , … , m s are positive integers with d ̲ ( φ ) = m 0 < m 1 < ⋯ < m s = d ( φ ) .

By using the same argument as used in Case 1, we know that b j ( z ) and d i ( z ) are small functions of f .

By Lemma 3, we obtain

(20) T ( r , φ ( f ) ) = d ( φ ) T ( r , f ) + S ( r , f ) .

It follows from (19), (20), (3), and Lemma 1 that

(21) T ( r , φ ( f ) ) ≤ N ¯ ( r , φ ( f ) ) + N ¯ r , 1 φ ( f ) + N ¯ r , 1 φ ( f ) − a + S ( r , φ ( f ) ) ≤ N ¯ r , 1 d 0 f m 0 + ⋯ + d s f m s + N ¯ r , 1 φ ( f ) − a + S ( r , φ ( f ) ) ≤ N ¯ r , 1 f m 0 ( d 0 + ⋯ + d s f m s − m 0 ) + N ¯ r , 1 φ ( f ) − a + S ( r , φ ( f ) ) ≤ ( m s − m 0 ) T ( r , f ) + N ¯ r , 1 φ ( f ) − a + S ( r , φ ( f ) ) = ( d ( φ ) − d ̲ ( φ ) ) T ( r , f ) + N ¯ r , 1 φ ( f ) − a + S ( r , φ ( f ) ) = d ( φ ) − d ̲ ( φ ) d ( φ ) T ( r , φ ( f ) ) + N ¯ r , 1 φ ( f ) − a + S ( r , φ ( f ) ) .

By (21), we have

d ̲ ( φ ) d ( φ ) T ( r , φ ( f ) ) ≤ N ¯ r , 1 φ ( f ) − a + S ( r , φ ( f ) ) .

It follows that

d ̲ ( φ ) d ( φ ) ≤ lim ¯ r → ∞ N ¯ r , 1 φ ( f ) − a T ( r , φ ( f ) ) + lim ̲ r → ∞ S ( r , φ ( f ) ) T ( r , φ ( f ) ) = lim ¯ r → ∞ N ¯ r , 1 φ ( f ) − a T ( r , φ ( f ) ) .

Hence, we have Θ ( a , φ ( f ) ) ≤ 1 − d ̲ ( φ ) d ( φ ) .

This completes the proof of Theorem 2.

Remark 1

In Theorem 2, if a ( z ) ≡ 0 , we have the following results.

If φ ( f ) is a homogeneous differential polynomial. Then, by (3), (16), and (17), we have

N ¯ r , 1 φ ( f ) = N ¯ r , 1 B f d ( φ ) = S ( r , f ) .

If φ ( f ) is a non-homogeneous differential polynomial. Then, by (3) and (19), we obtain

T ( r , f ) ≤ N ¯ r , 1 φ ( f ) + S ( r , f ) = N ¯ r , 1 d 0 f m 0 + d 1 f m 1 + ⋯ + d s f m s + S ( r , f ) = N ¯ r , 1 f m 0 ( d 0 + ⋯ + d s f m s − m 0 ) + S ( r , f ) ≤ ( m s − m 0 ) T ( r , f ) + S ( r , f ) = ( d ( φ ) − d ̲ ( φ ) ) T ( r , f ) + S ( r , f ) .

That is,

(22) T ( r , f ) ≤ N ¯ r , 1 φ ( f ) + S ( r , f ) ≤ ( d ( φ ) − d ̲ ( φ ) ) T ( r , f ) + S ( r , f ) .

Example 5

Let f = e z and φ ( f ) = f 4 − 2 f f ′ . Hence, we obtain

(23) N ¯ r , 1 φ ( f ) = N ¯ r , 1 e 2 z − 2 ≤ 2 T ( r , e z ) + S ( r , f ) .

By Nevanlinna’s second fundamental theorem, we have

(24) 2 T ( r , e z ) = T ( r , e 2 z ) ≤ N ¯ ( r , e 2 z ) + N ¯ r , 1 e 2 z + N ¯ r , 1 e 2 z − 2 + S ( r , f ) ≤ N ¯ r , 1 e 2 z − 2 + S ( r , f ) .

It follows from (23) and (24) that

N ¯ r , 1 φ ( f ) = 2 T ( r , e z ) + S ( r , f ) = ( d ( φ ) − d ̲ ( φ ) ) T ( r , f ) + S ( r , f ) .

This example shows that the bound ( d ( φ ) − d ̲ ( φ ) ) T ( r , f ) + S ( r , f ) in (22) is sharp.

Example 6

Let f = e z and φ ( f ) = f 3 − f ′ 2 . Then, by using the same argument as used in Example 5, we know that N ¯ r , 1 φ ( f ) = T ( r , f ) + S ( r , f ) . Hence, the bound T ( r , f ) + S ( r , f ) in (22) is sharp.

Remark 2

Condition (3) is replaced by N ¯ ( r , f ) + N ¯ r , 1 f = S ( r , f ) , Theorem 2 still holds.

Acknowledgments

We are very grateful to the anonymous referees for their careful review and valuable suggestions.

  1. Funding information: This work was supported by the National Natural Science Foundation of China (Grant No. 12171127) and the Natural Science Foundation of Zhejiang Province (Grant No. LY21A010012).

  2. Conflict of interest: The authors state that there is no conflict of interest.

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Received: 2022-05-28
Revised: 2022-11-25
Accepted: 2023-01-25
Published Online: 2023-03-14

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Special Issue on Future Directions of Further Developments in Mathematics
  2. What will the mathematics of tomorrow look like?
  3. On H 2-solutions for a Camassa-Holm type equation
  4. Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
  5. Control of multi-agent systems: Results, open problems, and applications
  6. Logical perspectives on the foundations of probability
  7. Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
  8. A non-smooth Brezis-Oswald uniqueness result
  9. Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
  10. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
  11. Positive solution for a nonlocal problem with strong singular nonlinearity
  12. Analysis of solutions for the fractional differential equation with Hadamard-type
  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
  17. On the use of L-functionals in regression models
  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
  20. Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
  25. On the uniqueness of limit cycles for generalized Liénard systems
  26. A derivative-Hilbert operator acting on Dirichlet spaces
  27. Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
  30. Some results on the value distribution of differential polynomials
  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
  32. Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
  33. Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
  34. Homotopy cartesian squares in extriangulated categories
  35. A unified perspective on some autocorrelation measures in different fields: A note
  36. Total Roman domination on the digraphs
  37. Well-posedness for bilevel vector equilibrium problems with variable domination structures
  38. Binet's second formula, Hermite's generalization, and two related identities
  39. Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients
  40. Multidimensional sampling-Kantorovich operators in BV-spaces
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  42. Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space
  43. Relating the super domination and 2-domination numbers in cactus graphs
  44. Compatibility of the method of brackets with classical integration rules
  45. On the inverse Collatz-Sinogowitz irregularity problem
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  47. Global analysis and control for a vector-borne epidemic model with multi-edge infection on complex networks
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  50. A comparison of some confidence intervals for a binomial proportion based on a shrinkage estimator
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  52. Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
  53. Mean square exponential stability of stochastic function differential equations in the G-framework
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  55. Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
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  60. On semigroups of transformations that preserve a double direction equivalence
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  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
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  67. On the Waring-Goldbach problem for two squares and four cubes
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  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
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  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
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  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
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  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
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  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
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  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
https://doi.org/10.1515/math-2022-0560
Keywords for this article
meromorphic function; small function; deficiency value
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Articles in the same Issue

  1. Special Issue on Future Directions of Further Developments in Mathematics
  2. What will the mathematics of tomorrow look like?
  3. On H 2-solutions for a Camassa-Holm type equation
  4. Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
  5. Control of multi-agent systems: Results, open problems, and applications
  6. Logical perspectives on the foundations of probability
  7. Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
  8. A non-smooth Brezis-Oswald uniqueness result
  9. Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
  10. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
  11. Positive solution for a nonlocal problem with strong singular nonlinearity
  12. Analysis of solutions for the fractional differential equation with Hadamard-type
  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
  17. On the use of L-functionals in regression models
  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
  20. Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
  25. On the uniqueness of limit cycles for generalized Liénard systems
  26. A derivative-Hilbert operator acting on Dirichlet spaces
  27. Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
  30. Some results on the value distribution of differential polynomials
  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
  32. Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
  33. Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
  34. Homotopy cartesian squares in extriangulated categories
  35. A unified perspective on some autocorrelation measures in different fields: A note
  36. Total Roman domination on the digraphs
  37. Well-posedness for bilevel vector equilibrium problems with variable domination structures
  38. Binet's second formula, Hermite's generalization, and two related identities
  39. Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients
  40. Multidimensional sampling-Kantorovich operators in BV-spaces
  41. A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems
  42. Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space
  43. Relating the super domination and 2-domination numbers in cactus graphs
  44. Compatibility of the method of brackets with classical integration rules
  45. On the inverse Collatz-Sinogowitz irregularity problem
  46. Positive solutions for boundary value problems of a class of second-order differential equation system
  47. Global analysis and control for a vector-borne epidemic model with multi-edge infection on complex networks
  48. Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities
  49. On 2r-ideals in commutative rings with zero-divisors
  50. A comparison of some confidence intervals for a binomial proportion based on a shrinkage estimator
  51. The construction of nuclei for normal constituents of Bπ-characters
  52. Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
  53. Mean square exponential stability of stochastic function differential equations in the G-framework
  54. Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
  55. Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
  61. Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
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