Home Some results on value distribution concerning Hayman's alternative
Article Open Access

Some results on value distribution concerning Hayman's alternative

  • Xinyu Zhuang , Ruilin Zheng , Zhiying He and Mingliang Fang EMAIL logo
Published/Copyright: February 4, 2025
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 23 Issue 1

Abstract

In this article, we study the value distribution of meromorphic functions concerning Hayman’s alternative. We extend and improve some results due to Zhu [A far-reaching form of Hayman’s inequality; fixed points of meromorphic functions and their derivatives, Kexue Tongbao (Chinese) 31 (1986), no. 11, 801–804], Hua and Chuang [On a conjecture of Hayman, Acta Math. Sinica (N.S.) 7 (1991), no. 2, 119–126], and Charak and Singh [A value distribution result related to Hayman’s alternative, Commun. Korean Math. Soc. 34 (2019), no. 2, 495–506].

Keywords: meromorphic functions; value distribution; differential polynomials; Hayman’s alternative
MSC 2010: 30D30; 30D35

1 Introduction and main results

In this article, we assume that the reader is familiar with the basic notions of Nevanlinna’s value distribution theory [1–4]. In the following, a meromorphic function always means meromorphic in the whole complex plane. By S ( r , f ) , we denote any quantity satisfying S ( r , f ) = o ( T ( r , f ) ) as r → ∞ possible outside of an exceptional set E with finite measure.

Let f be a meromorphic function, and let k be a positive integer. We denote by N k ) ( r , f ) the counting function for poles of f with multiplicity ≤ k , counting multiplicity and by N ¯ k ) ( r , f ) the corresponding one for which multiplicity is not counted. Let N ( k ( r , f ) be the counting function for poles of f with multiplicity ≥ k , counting multiplicity and by N ¯ ( k ( r , f ) be the corresponding one for which multiplicity is not counted.

Let f be a nonconstant meromorphic function, let a 1 , a 2 , … , a k be small functions of f , and let n 0 , n 1 , n 2 , … , n k be nonnegative integers. We define the differential monomial of f as follows:

M ( f ) = f n 0 ( f ′ ) n 1 … ( f ( k ) ) n k ,

and its degree γ M = n 0 + n 1 + … + n k . Let M 1 ( f ) , M 2 ( f ) , … , M n ( f ) be differential monomials. We define the differential polynomial of f as follows:

(1.1) H ( f ) = a 1 M 1 ( f ) + a 2 M 2 ( f ) + … + a n M n ( f ) ,

and define γ H = max { γ M 1 , γ M 2 , … , γ M n } , γ ̲ H = min { γ M 1 , γ M 2 , … , γ M n } by the degree and the lower degree of H , respectively.

In 1986, Zhu [5] proved the following result.

Theorem 1.1

Let f be a transcendental meromorphic function, and let φ ( ≢ 0 ) be a small function of f. Then

T ( r , f ) ≤ 8 N r , 1 f + 8 N ¯ r , 1 f ′ − φ + S ( r , f ) .

In this article, we improve Theorem 1.1 as follows.

Theorem 1.2

Let f be a transcendental meromorphic function, and let φ ( ≢ 0 ) be a small function of f. Then

T ( r , f ) ≤ 5 N r , 1 f + 5 N ¯ r , 1 f ′ − φ + S ( r , f ) .

In 1991, Hua and Chuang [6] proved the following result.

Theorem 1.3

Let f be a nonconstant meromorphic function, and let n , m be two positive integers. Assume that Q ( f ) = f m H ( f ) , where H ( f ) ( ≢ 0 ) is a differential polynomial defined by (1.1). Then for any nonzero complex number b :

  1. If n ≥ 3 , then T ( r , f ) ≤ 2 m N ¯ r , 1 Q n Q ′ − b + S ( r , f ) ;

  2. If n = 2 , then T ( r , f ) ≤ 1 m N ¯ ( r , f ) + 2 m N ¯ r , 1 Q 2 Q ′ − b + S ( r , f ) .

In this article, we prove the following result.

Theorem 1.4

Let f be a nonconstant meromorphic function, and let n , m be two positive integers. Assume that Q ( f ) = f m H ( f ) , where H ( f ) ( ≢ 0 ) is a differential polynomial defined by (1.1). Then for any nonzero complex number b ,

  1. If n ≥ 3 , then T ( r , f ) ≤ 1 m N ¯ r , 1 Q n Q ′ − b + S ( r , f ) ;

  2. If n = 2 , then T ( r , f ) ≤ 1 m N ¯ ( r , f ) + 1 m N ¯ r , 1 Q 2 Q ′ − b + S ( r , f ) .

In 2019, Charak and Singh [7] proved the following result.

Theorem 1.5

Let f be a transcendental meromorphic function, let ϕ be a small function of f such that f and ϕ have no common poles, and let k be a positive integer. If f ≠ 0 and f ( k ) ≠ ϕ , then f ( k + 1 ) = ϕ and f ( k + 1 ) = ϕ ′ have infinitely many solutions.

Remark 1.6

Theorem 1.5 is not valid by the following example.

Example 1.7

Let f = e z , and let ϕ ≡ 0 . Obviously, f ≠ 0 and f ( k ) ≠ ϕ , but f ( k + 1 ) = ϕ does not have infinitely many solutions.

Although Theorem 1.5 is not valid, we have the following result.

Theorem 1.8

Let f be a transcendental meromorphic function, let ϕ be a small function of f, and let k be a positive integer. If f ≠ 0 and f ( k ) ≠ ϕ , then ϕ ≡ 0 . In addition, if k ≥ 2 , then f = e a z + b , where a ( ≠ 0 ) , b are constants; if k = 1 , then f ″ has infinitely many zeros, except f = e a z + b , where a ( ≠ 0 ) , b are constants.

Remark 1.9

f ≠ 0 refers to the fact that for any z ∈ C , it holds that f ( z ) ≠ 0 . f ( k ) ≠ ϕ refers to the fact that for any z ∈ C , it holds that f ( k ) ( z ) ≠ ϕ ( z ) .

The following examples show that two cases occur in Theorem 1.8.

Example 1.10

Let f ( z ) = e z , and let k be a positive integer. Obviously, f ≠ 0 , f ( k ) ≠ 0 .

Example 1.11

Let f ( z ) = e e z . Obviously, f ≠ 0 , f ′ ≠ 0 . We have f ″ = e e z e z ( e z + 1 ) . Thus, f ″ has infinitely many zeros.

2 Some lemmas

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

Lemma 2.1

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

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

Lemma 2.2

[1,3] Let f be a nonconstant meromorphic function, let n be a positive integer, and let a 1 , a 2 , … , a n be distinct small functions of f. Then

m r , 1 f − a 1 + … + m r , 1 f − a n ≤ m r , 1 f − a 1 + … + 1 f − a n + S ( r , f ) .

It follows from the theorem in [8, p. 247] the following result.

Lemma 2.3

Let f be a transcendental meromorphic function. Then

1 4 ≤ T ( r , f ) T ( r , f ′ ) ≤ 3 e + 1 4 ,

as r → ∞ on a set of positive lower logarithmic density.

Lemma 2.4

[1,3] Let f 1 and f 2 be two nonconstant meromorphic functions. Then

N ( r , f 1 f 2 ) − N r , 1 f 1 f 2 = N ( r , f 1 ) + N ( r , f 2 ) − N r , 1 f 1 − N r , 1 f 2 .

Lemma 2.5

Let f be a nonconstant meromorphic function, let b be a nonzero complex number, and let n be a positive integer. Then we have

  1. n ≥ 3 , T ( r , f ) ≤ N ¯ r , 1 f n f ′ − b + S ( r , f ) ;

  2. n = 2 , T ( r , f ) ≤ N ¯ ( r , f ) + N ¯ r , 1 f 2 f ′ − b + S ( r , f ) .

Proof

By Lemmas 2.1 and 2.2 and Nevanlinna’s first fundamental theorem, we have

(2.1) m r , 1 f n + 1 + m r , 1 f n f ′ − b ≤ m r , f n f ′ f n + 1 + m r , 1 f n f ′ + m r , 1 f n f ′ − b ≤ m r , 1 f n f ′ + 1 f n f ′ − b + S ( r , f ) ≤ m r , ( f n f ′ ) ′ f n f ′ + ( f n f ′ − b ) ′ f n f ′ − b + m r , 1 ( f n f ′ ) ′ + S ( r , f ) ≤ T ( r , ( f n f ′ ) ′ ) − N r , 1 ( f n f ′ ) ′ + S ( r , f ) ≤ T ( r , f n f ′ ) + N ¯ ( r , f ) − N r , 1 ( f n f ′ ) ′ + S ( r , f ) .

Then, by adding N r , 1 f n + 1 + N r , 1 f n f ′ − b to both sides of (2.1), we have

m r , 1 f n + 1 + m r , 1 f n f ′ − b + N r , 1 f n + 1 + N r , 1 f n f ′ − b ≤ N r , 1 f n + 1 + N r , 1 f n f ′ − b + T ( r , f n f ′ ) + N ¯ ( r , f ) − N r , 1 ( f n f ′ ) ′ + S ( r , f ) .

It follows

T r , 1 f n + 1 + T r , 1 f n f ′ − b ≤ N r , 1 f n + 1 + N r , 1 f n f ′ − b + T ( r , f n f ′ ) + N ¯ ( r , f ) − N r , 1 ( f n f ′ ) ′ + S ( r , f ) .

By Nevanlinna’s first fundamental theorem, we have

T r , 1 f n f ′ − b = T ( r , f n f ′ ) + O ( 1 ) .

Thus, we obtain

T r , 1 f n + 1 ≤ N r , 1 f n + 1 + N r , 1 f n f ′ − b + N ¯ ( r , f ) − N r , 1 ( f n f ′ ) ′ + S ( r , f ) .

From T r , 1 f n + 1 = ( n + 1 ) T ( r , f ) + S ( r , f ) , we obtain

( n + 1 ) T ( r , f ) ≤ N r , 1 f n + 1 + N r , 1 f n f ′ − b + N ¯ ( r , f ) − N r , 1 ( f n f ′ ) ′ + S ( r , f ) .

If z 0 is a zero of f with multiplicity l 1 , then z 0 is a zero of f n + 1 with multiplicity ( n + 1 ) l 1 . Hence, z 0 must be a zero of ( f n f ′ ) ′ with multiplicity ( n + 1 ) l 1 − 2 . Similarly, if z 0 is a zero of f n f ′ − b with multiplicity l 2 , then z 0 must be a zero of ( f n f ′ − b ) ′ with multiplicity l 2 − 1 , which yields z 0 is also a zero of ( f n f ′ ) ′ with multiplicity l 2 − 1 . It follows

N r , 1 f n + 1 + N r , 1 f n f ′ − b − N r , 1 ( f n f ′ ) ′ ≤ 2 N ¯ r , 1 f + N ¯ r , 1 f n f ′ − b − N 0 * r , 1 ( f n f ′ ) ′ ,

where N 0 * r , 1 ( f n f ′ ) ′ is the counting function for the zeros of ( f n f ′ ) ′ , which are not zeros of f n + 1 ( f n f ′ − b ) .

Hence, we have

(2.2) ( n + 1 ) T ( r , f ) ≤ N r , 1 f n + 1 + N r , 1 f n f ′ − b + N ¯ ( r , f ) − N r , 1 ( f n f ′ ) ′ + S ( r , f ) ≤ N ¯ ( r , f ) + 2 N ¯ r , 1 f + N ¯ r , 1 f n f ′ − b − N 0 * r , 1 ( f n f ′ ) ′ + S ( r , f ) ≤ N ¯ ( r , f ) + 2 N ¯ r , 1 f + N ¯ r , 1 f n f ′ − b + S ( r , f ) .

Next we consider two cases.

Case 1. n ≥ 3 . From (2.2), we obtain

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

It follows

(2.3) T ( r , f ) ≤ N ¯ r , 1 f n f ′ − b + S ( r , f ) .

Case 2. n = 2 . From (2.2), we have

(2.4)□ T ( r , f ) ≤ N ¯ ( r , f ) + N ¯ r , 1 f 2 f ′ − b + S ( r , f ) .

Lemma 2.6

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

N r , 1 f ( k ) ≤ N r , 1 f + k N ¯ ( r , f ) + S ( r , f ) .

Lemma 2.7

[9] Let f be a transcendental meromorphic function, let ϕ ( ≢ 0 ) be a small function of f, and let k be a positive integer. If N r , 1 f = S ( r , f ) , then N r , 1 f ( k ) − ϕ ≠ S ( r , f ) .

Lemma 2.8

[10] Let f be a nonconstant meromorphic function, and let k ( ≥ 2 ) be an integer. If f ≠ 0 , f ( k ) ≠ 0 , then either f = e a z + b or f = ( a z + b ) − n , where a ( ≠ 0 ) , b are constants and n is a positive integer.

Remark 2.9

f ≠ 0 refer to the fact that for any z ∈ C , it holds that f ( z ) ≠ 0 . f ( k ) ≠ 0 refer to the fact that for any z ∈ C , it holds that f ( k ) ( z ) ≠ 0 .

It follows from Theorem 2.2 in [11, p. 423] the following result.

Lemma 2.10

Let f be a nonconstant meromorphic function, and let k ( ≥ 2 ) be an integer. If f and f ( k ) have finitely many zeros, then f = R e P , where R is a rational function and P is a polynomial.

3 Proof of Theorem 1.2

Set L = f ′ φ f ″ φ ′ . By Lemmas 2.1 and 2.2, we obtain

(3.1) m r , 1 f + m r , 1 f ′ − φ ≤ m r , f ′ f + m r , 1 f ′ + m r , 1 f ′ − φ ≤ m r , 1 f ′ + 1 f ′ − φ + S ( r , f ) ≤ m r , L f ′ + L f ′ − φ + m r , 1 L + S ( r , f ) ≤ m r , L f ′ + m r , L f ′ − φ + m r , 1 L + S ( r , f ) .

From the definition of L , φ ( ≢ 0 ) is a small function of f , and from Lemma 2.1, we have

(3.2) m r , L f ′ = m r , φ ′ − φ f ″ f ′ ≤ m ( r , φ ′ ) + m ( r , φ ) + m r , f ″ f ′ + S ( r , f ) ≤ S ( r , f )

and

(3.3) m r , L f ′ − φ = m r , φ ′ − φ ( f ′ − φ ) ′ f ′ − φ ≤ m ( r , φ ′ ) + m ( r , φ ) + m r , ( f ′ − φ ) ′ f ′ − φ + S ( r , f ) ≤ S ( r , f ) .

By combining (3.1)–(3.3) and Nevanlinna’s first fundamental theorem, we obtain

(3.4) m r , 1 f + m r , 1 f ′ − φ ≤ m r , 1 L + S ( r , f ) ≤ T r , 1 L − N r , 1 L + S ( r , f ) ≤ T ( r , L ) − N r , 1 L + S ( r , f ) .

From the definition of L and (3.3), we have

(3.5) m ( r , L ) = m r , ( f ′ − φ ) φ ′ − φ f ″ − φ ′ f ′ − φ ≤ m ( r , f ′ − φ ) + m r , φ ′ − φ f ″ − φ ′ f ′ − φ + S ( r , f ) ≤ m ( r , f ′ − φ ) + S ( r , f ) .

Similarly,

(3.6) N ( r , L ) = N ( r , ( f ′ − φ ) φ ′ − ( f ″ − φ ′ ) φ ) ≤ N ( r , ( f ′ − φ ) ′ ) + S ( r , f ) ≤ N ( r , f ′ − φ ) + N ¯ ( r , f ′ − φ ) + S ( r , f ) ≤ N ( r , f ′ − φ ) + N ¯ ( r , f ) + S ( r , f ) .

By combining (3.4)–(3.6), we obtain

(3.7) m r , 1 f + m r , 1 f ′ − φ ≤ T ( r , L ) − N r , 1 L + S ( r , f ) ≤ m ( r , L ) + N ( r , L ) − N r , 1 L + S ( r , f ) ≤ m ( r , f ′ − φ ) + N ( r , f ′ − φ ) + N ¯ ( r , f ) − N r , 1 L + S ( r , f ) ≤ T ( r , f ′ − φ ) + N ¯ ( r , f ) − N r , 1 L + S ( r , f ) .

By adding N r , 1 f + N r , 1 f ′ − φ to both sides of (3.7), we have

m r , 1 f + m r , 1 f ′ − φ + N r , 1 f + N r , 1 f ′ − φ ≤ N r , 1 f + N r , 1 f ′ − φ + T ( r , f ′ − φ ) + N ¯ ( r , f ) − N r , 1 L + S ( r , f ) .

Hence, we obtain

T r , 1 f + T r , 1 f ′ − φ ≤ N r , 1 f + N r , 1 f ′ − φ + T ( r , f ′ − φ ) + N ¯ ( r , f ) − N r , 1 L + S ( r , f ) .

It follows from Nevanlinna’s first fundamental theorem that

T r , 1 f = T ( r , f ) + O ( 1 ) , T r , 1 f ′ − φ = T ( r , f ′ − φ ) + O ( 1 ) .

Therefore,

(3.8) T ( r , f ) ≤ N r , 1 f + N r , 1 f ′ − φ + N ¯ ( r , f ) − N r , 1 L + S ( r , f ) ≤ N ¯ ( r , f ) + N r , 1 f + N ¯ r , 1 f ′ − φ + S ( r , f ) .

It follows from (3.8) and N ¯ ( r , f ) = N ¯ 1 ) ( r , f ) + N ¯ ( 2 ( r , f ) that

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

Hence, we obtain

(3.9) T ( r , f ) ≤ N 1 ) ( r , f ) + 2 N r , 1 f + 2 N ¯ r , 1 f ′ − φ + S ( r , f ) .

Obviously, f ″ φ − φ ′ f ′ ≢ 0 . Set

P = f ( f ′ − φ ) f ″ φ − φ ′ f ′ , G = ( P φ ) ″ + ( P φ ′ ) ′ + φ .

Thus, we have

(3.10) P φ f ″ − f ( f ′ − φ ) ≡ P φ ′ f ′ .

Let z 0 be a simple pole of f , and satisfy that φ ( z 0 ) ≠ 0 , ∞ , φ ′ ( z 0 ) ≠ 0 . Obviously, P φ and P φ ′ are holomorphic at z 0 , so we obtain

(3.11) f ( z ) = c − 1 z − z 0 + c 0 + c 1 ( z − z 0 ) + c 2 ( z − z 0 ) 2 + … ,

where c − 1 ( ≠ 0 ) , c 0 , c 1 , c 2 , … are constants.

(3.12) φ ( z ) = φ ( z 0 ) + φ ′ ( z 0 ) ( z − z 0 ) + 1 2 φ ″ ( z 0 ) ( z − z 0 ) 2 + … ,

(3.13) P φ ( z ) = P φ ( z 0 ) + ( P φ ) ′ ( z 0 ) ( z − z 0 ) + 1 2 ( P φ ) ″ ( z 0 ) ( z − z 0 ) 2 + … ,

(3.14) P φ ′ ( z ) = P φ ′ ( z 0 ) + ( P φ ′ ) ′ ( z 0 ) ( z − z 0 ) + 1 2 ( P φ ′ ) ″ ( z 0 ) ( z − z 0 ) 2 + … .

By substituting (3.11)–(3.14) into (3.10), and comparing the coefficients of 1 z − z 0 , we obtain

( P φ ) ″ ( z 0 ) + ( P φ ′ ) ′ ( z 0 ) + φ ( z 0 ) = 0 .

That is, G ( z 0 ) = 0 .

Next we consider the following two cases.

Case 1. G ≢ 0 . Since φ is a small function of f , then by Nevanlinna’s first fundamental theorem and Lemma 2.1, we have

(3.15) N 1 ) ( r , f ) ≤ N r , 1 G + N ¯ r , 1 φ + N ¯ r , 1 φ ′ + N ¯ ( r , φ ) ≤ N r , 1 G + S ( r , f ) ≤ T ( r , G ) + S ( r , f ) ≤ m ( r , P ″ φ + 3 P ′ φ ′ + 2 P φ ″ ) + m ( r , φ ) + N ( r , P ″ φ + 3 P ′ φ ′ + 2 P φ ″ ) + N ( r , φ ) + S ( r , f ) ≤ m ( r , P ) + m r , P ″ φ + 3 P ′ φ ′ + 2 P φ ″ P + N ( r , P ″ ) + S ( r , f ) ≤ m ( r , P ) + N ( r , P ) + 2 N ¯ ( r , P ) + S ( r , f ) ≤ 3 T ( r , P ) + S ( r , f ) .

From Lemma 2.1, we have

(3.16) m r , 1 P = m r , f ″ φ − φ ′ f ′ f ( f ′ − φ ) ≤ m r , f ″ φ − φ ′ f ′ f ′ ( f ′ − φ ) + m r , f ′ f + S ( r , f ) = m r , f ′ − φ f ′ ′ f ′ − φ f ′ + m r , f ′ f + S ( r , f ) ≤ S ( r , f ) .

From the definition of P , we have

N r , 1 P = N r , f ″ φ − φ ′ f ′ f ( f ′ − φ ) .

Since φ ( ≢ 0 ) is a small function of f , we have T ( r , φ ) = S ( r , f ) , it follows S ( r , φ ) = S ( r , f ) . Hence, we obtain T ( r , φ ′ ) = S ( r , f ) .

Next we consider the following three subcases.

Case 1.1. Let z 0 be a pole of f with multiplicity l 1 ( ≥ 1 ) , but φ ( z 0 ) ≠ ∞ . It follows that z 0 is a pole of f ′ − φ and f ″ φ − φ ′ f ′ with multiplicity l 1 + 1 and l 1 + 2 , respectively. In this case, the multiplicity of the poles in the numerator is not greater than that in the denominator. Hence, z 0 is not a pole of 1 P .

Case 1.2. Let z 0 be a pole of f with multiplicity l 1 ( ≥ 1 ) and φ ( z 0 ) = ∞ . From N ( r , φ ) = S ( r , f ) , we can deduce that the zeros of this type of P is S ( r , f ) .

Case 1.3. Let z 0 be a zero of f ′ − φ with multiplicity l 2 ( ≥ 1 ) , but not the zero of f . It follows that z 0 is zero of f ″ − φ ′ with multiplicity l 2 − 1 . From

f ″ φ − φ ′ f ′ = f ″ φ − φ φ ′ + φ φ ′ − φ ′ f ′ = φ ( f ″ − φ ′ ) − φ ′ ( f ′ − φ ) ,

we have

1 P = f ″ φ − φ ′ f ′ f ( f ′ − φ ) = φ ( f ″ − φ ′ ) − φ ′ ( f ′ − φ ) f ( f ′ − φ ) .

It follows that z 0 is a simple pole of 1 P .

Thus, we have

(3.17) N r , 1 P = N r , f ″ φ − φ ′ f ′ f ( f ′ − φ ) + S ( r , f ) ≤ N r , 1 f + N ¯ r , 1 f ′ − φ + S ( r , f ) .

By (3.15)–(3.17), and Nevanlinna’s first fundamental theorem, we obtain

(3.18) N 1 ) ( r , f ) ≤ 3 N r , 1 f + 3 N ¯ r , 1 f ′ − φ + S ( r , f ) .

Case 2. G ≡ 0 . That is,

(3.19) P ″ φ + 3 P ′ φ ′ + 2 P φ ″ + φ ≡ 0 .

Let z 0 be a pole of P with multiplicity l . Then z 0 must be either a zero or a pole of φ .

In fact, if φ ( z 0 ) = a , where a ( ≠ 0 , ∞ ) is a constant, then z 0 is a pole of P ″ φ with multiplicity l + 2 , a pole of P ′ φ ′ with multiplicity at most l + 1 , and a pole of P φ ″ with multiplicity at most l . Thus, by (3.19), we know that z 0 is a pole of P ″ φ + 3 P ′ φ ′ + 2 P φ ″ + φ , a contradiction.

Hence,

N ¯ ( r , P ) ≤ N ¯ ( r , φ ) + N ¯ r , 1 φ ≤ S ( r , f ) .

We define N 0 * r , 1 f ″ φ − φ ′ f ′ is the counting function for the zeros of f ″ φ − φ ′ f ′ , which are not zeros of f ( f ′ − φ ) , and N ¯ 0 * r , 1 f ″ φ − φ ′ f ′ as the corresponding function where the multiplicity is not counted.

Let z 0 be a zero of f ″ φ − φ ′ f ′ , but not the zero of f ( f ′ − φ ) . Then z 0 is a pole of P . It follows from N ¯ ( r , P ) ≤ S ( r , f ) that

(3.20) N ¯ 0 * r , 1 f ″ φ − φ ′ f ′ ≤ N ¯ ( r , P ) ≤ S ( r , f ) .

Let z 1 be a pole of f with multiplicity l 1 ( ≥ 2 ) , but φ ( z 1 ) ≠ 0 , ∞ . Obviously, z 1 is a pole of f ( f ′ − φ ) and f ″ φ − φ ′ f ′ with multiplicity 2 l 1 + 1 and l 1 + 2 , respectively. From the definition of P , we deduce that z 1 must be a pole of P with multiplicity l 1 − 1 . Hence, we have

(3.21) N ¯ ( 2 ( r , f ) ≤ N ¯ ( r , P ) + N ¯ ( r , φ ) + N ¯ r , 1 φ ≤ S ( r , f ) .

Set

g = φ 3 ( f ′ − φ ) 3 ( f ″ φ − f ′ φ ′ ) 2 = φ ( f ′ − φ ) f ″ − φ ′ f ′ − φ − φ ′ φ 2 .

Let z 0 be a simple pole of f with φ ( z 0 ) ≠ 0 , ∞ , φ ′ ( z 0 ) ≠ 0 . Then by (3.11) and (3.12), we obtain

g = − φ ( z 0 ) c − 1 4 + λ 2 ( z − z 0 ) 2 + λ 3 ( z − z 0 ) 3 + … ,

where − φ ( z 0 ) c − 1 4 ≠ 0 , λ 2 , λ 3 , … are constants. Hence, g ( z 0 ) ≠ 0 , ∞ , and g ′ ( z 0 ) = 0 .

Next we consider two subcases.

Case 2.1. g ′ ≢ 0 . It follows from that φ is a small function of f that

(3.22) N 1 ) ( r , f ) ≤ N 0 r , 1 g ′ + N ¯ r , 1 φ + N ¯ r , 1 φ ′ + N ¯ ( r , φ ) ≤ N 0 r , 1 g ′ + S ( r , f ) ,

where N 0 r , 1 g ′ is the counting function for the zeros of g ′ , which are not the zeros of g .

By Nevanlinna’s first fundamental theorem and Lemma 2.1, we have

(3.23) N r , g g ′ − N r , g ′ g = T r , g g ′ − m r , g g ′ − T r , g ′ g + m r , g ′ g = − m r , g g ′ + S ( r , g ) + S ( r , f ) = − m r , g g ′ + S ( r , f ) .

Referring to line -8 and line -6 in [1, p. 57], we obtain

(3.24) N ( r , g ′ ) − N ( r , g ) = N ¯ ( r , g )

and

(3.25) N r , 1 g − N r , 1 g ′ = N ¯ r , 1 g − N 0 r , 1 g ′ .

From (3.24), (3.25), and Lemma 2.4, we have

(3.26) N r , g g ′ − N r , g ′ g = N ( r , g ) + N r , 1 g ′ − N ( r , g ′ ) − N r , 1 g = N 0 r , 1 g ′ − N ¯ r , 1 g − N ¯ ( r , g ) .

By (3.20), (3.21), and the definition of g , we have

(3.27) N ¯ ( r , g ) + N ¯ r , 1 g ≤ N ¯ ( 2 ( r , f ) + N ¯ r , 1 f + N ¯ r , 1 f ′ − φ + N ¯ 0 * r , 1 f ″ φ − φ ′ f ′ + S ( r , f ) ≤ N ¯ r , 1 f + N ¯ r , 1 f ′ − φ + S ( r , f ) .

Combining (3.22), (3.23), (3.26) with (3.27), we obtain

(3.28) N 1 ) ( r , f ) ≤ N 0 r , 1 g ′ + S ( r , f ) ≤ N ¯ r , 1 g + N ¯ ( r , g ) − m r , g g ′ + S ( r , f ) ≤ N ¯ r , 1 g + N ¯ ( r , g ) + S ( r , f ) ≤ N ¯ r , 1 f + N ¯ r , 1 f ′ − φ + S ( r , f ) .

Case 2.2. g ′ ≡ 0 . Then g ≡ A , where A is a constant.

If A = 0 , then φ ( f ′ − φ ) f ″ − φ ′ f ′ − φ − φ ′ φ 2 ≡ 0 . Hence, we obtain that either φ ≡ 0 or f ′ − φ ≡ 0 , a contradiction. Thus, A ≠ 0 . Hence, we have

(3.29) φ ( f ′ − φ ) ≡ A f ″ − φ ′ f ′ − φ − φ ′ φ 2 .

Set F = f ′ − φ φ . Then by (3.29), we obtain

(3.30) F 3 = A φ 2 ( F ′ ) 2 .

Obviously, F ≢ 0 . Thus, we have 1 F = A φ 2 1 F ′ 2 .

Set H = 1 F . Then we have H = B H ′ φ 2 , where B 2 = A ≠ 0 . Hence, H = h 2 , H ′ = 2 h h ′ , where h is a meromorphic function. It follows ( h ′ ) 2 = φ 2 B 2 . Hence, T ( r , h ′ ) = S ( r , f ) .

Since h 2 = φ f ′ − φ , we have

(3.31) 2 T ( r , h ) = T r , φ f ′ − φ = T ( r , f ′ ) + S ( r , f ) .

From (3.31), Lemma 2.3, and T ( r , h ′ ) = S ( r , f ) , we deduce that T ( r , f ′ ) ≤ S ( r , f ) , a contradiction.

By (3.18), (3.28), and (3.9), we obtain

T ( r , f ) ≤ 5 N r , 1 f + 5 N ¯ r , 1 f ′ − φ + S ( r , f ) .

This completes the proof of Theorem 1.2.

4 Proof of Theorem 1.4

Obviously,

(4.1) m r , 1 f γ Q ≤ m r , Q f γ Q + m r , 1 Q + S ( r , f ) ,

where γ Q = γ Q ( f ) = m + γ H .

By (4.1) and Nevanlinna’s first fundamental theorem, we have

(4.2) T ( r , f γ Q ) ≤ T ( r , Q ) + N r , 1 f γ Q − N r , 1 Q + m r , Q f γ Q + S ( r , f ) .

Since ( m + γ H ) T ( r , f ) = T ( r , f γ Q ) + S ( r , f ) , then by (4.2) and Lemma 2.1, we obtain

( m + γ H ) T ( r , f ) ≤ T ( r , Q ) + N r , 1 f γ Q − N r , 1 Q + m r , Q f γ Q + S ( r , f ) ≤ T ( r , Q ) + ( m + γ H ) N r , 1 f − N r , 1 Q + ( γ H − γ ̲ H ) m r , 1 f + S ( r , f ) ≤ T ( r , Q ) + ( m + γ H ) N r , 1 f − N r , 1 Q + ( γ H − γ ̲ H ) T r , 1 f − N r , 1 f + S ( r , f ) ≤ T ( r , Q ) + ( m + γ ̲ H ) N r , 1 f − N r , 1 Q + ( γ H − γ ̲ H ) T ( r , f ) + S ( r , f ) .

Thus, we have

( m + γ ̲ H ) T ( r , f ) ≤ T ( r , Q ) + ( m + γ ̲ H ) N r , 1 f − N r , 1 Q + S ( r , f ) ≤ T ( r , Q ) + ( m + γ ̲ H ) N r , 1 f − m N r , 1 f + S ( r , f ) ≤ T ( r , Q ) + γ ̲ H N r , 1 f + S ( r , f ) .

It follows

(4.3) T ( r , f ) ≤ 1 m T ( r , Q ) + S ( r , f ) .

By (4.3) and Lemma 2.5, we prove Theorem 1.4.

5 Proof of Theorem 1.8

Suppose ϕ ≢ 0 . By Lemma 2.7, we have N r , 1 f ( k ) − ϕ ≠ S ( r , f ) , a contradiction. Hence, ϕ ≡ 0 .

Next we consider two cases.

Case 1. k ≥ 2 .

Since f is a transcendental meromorphic function, then by Lemma 2.8, we deduce that f = e a z + b , where a ( ≠ 0 ) , b are constants.

Case 2. k = 1 .

Suppose that f ″ has finitely many zeros. Then by Lemma 2.10 and f ≠ 0 , we deduce that f = 1 Q e P , where P , Q are polynomials. It follows from f ′ ≠ 0 that

(5.1) f ′ = e P ( P ′ Q − Q ′ ) Q 2 ≠ 0 .

Obviously, P ′ Q − Q ′ ≢ 0 and P ′ ≢ 0 .

Next we consider two subcases.

Case 2.1. There exists z such that P ′ ( z ) Q ( z ) − Q ′ ( z ) = 0 .

Let z 1 be a zero of P ′ ( z ) Q ( z ) − Q ′ ( z ) . By (5.1), we have Q ( z 1 ) = 0 . Hence, Q ′ ( z 1 ) = 0 .

Thus, we obtain

(5.2) P ′ ( z ) Q ( z ) − Q ′ ( z ) = ( z − z 1 ) l 1 φ 1 ( z ) ,

and

(5.3) Q ( z ) = ( z − z 1 ) l 2 φ 2 ( z ) ,

where l 1 , l 2 ( ≥ 2 ) are positive integers and φ 1 ( z ) , φ 2 ( z ) are two polynomials with φ 1 ( z 1 ) ≠ 0 , φ 2 ( z 1 ) ≠ 0 . It follows from (5.3) that

(5.4) Q ′ ( z ) = ( z − z 1 ) l 2 − 1 φ 3 ( z ) ,

where φ 3 ( z ) = l 2 φ 2 ( z ) + ( z − z 1 ) φ ′ ( z ) . Obviously, φ 3 ( z 1 ) = l 2 φ 2 ( z 1 ) ≠ 0 .

By (5.3) and (5.4), we have

(5.5) P ′ ( z ) Q ( z ) − Q ′ ( z ) = P ′ ( z ) ( z − z 1 ) l 2 φ 2 ( z ) − ( z − z 1 ) l 2 − 1 φ 3 ( z ) = ( z − z 1 ) l 2 − 1 [ P ′ ( z ) ( z − z 1 ) φ 2 ( z ) − φ 3 ( z ) ] = ( z − z 1 ) l 2 − 1 φ 4 ( z ) ,

where φ 4 ( z ) = P ′ ( z ) ( z − z 1 ) φ 2 ( z ) − φ 3 ( z ) . Obviously, φ 4 ( z 1 ) = − φ 3 ( z 1 ) ≠ 0 .

It follows from (5.2) and (5.5) that l 1 = l 2 − 1 . Then by (5.2) and (5.4), we know that z 1 is a zero of both P ′ ( z ) Q ( z ) − Q ′ ( z ) and Q ′ ( z ) with the same multiplicities.

Suppose that the distinct zeros of P ′ ( z ) Q ( z ) − Q ′ ( z ) are z 1 , z 2 , … , z s with multiplicities are m 1 , m 2 , … , m s , where s , m 1 , m 2 , … , m s are positive integers. It follows

P ′ ( z ) Q ( z ) − Q ′ ( z ) = A ( z − z 1 ) m 1 ( z − z 2 ) m 2 … ( z − z s ) m s ,

where A is a nonzero constant and m 1 + m 2 + … + m s = deg ( P ′ Q − Q ′ ) = deg P ′ + deg Q ≥ deg Q . Furthermore, we have

Q ′ ( z ) = ( z − z 1 ) m 1 ( z − z 2 ) m 2 … ( z − z s ) m s φ 5 ( z ) ,

where φ 5 ( z ) is a polynomial. Thus, we obtain deg Q ′ ≥ m 1 + m 2 + … + m s ≥ deg Q . Then we deduce that Q is a nonzero constant.

Since f = 1 Q e P and f ′ ≠ 0 , we obtain f ′ = 1 Q e P P ′ ≠ 0 . Thus, we obtain P ( z ) = a z + b 1 , where a ( ≠ 0 ) , b 1 are constants. It follows that f = e a z + b , where b is a constant.

Case 2.2. P ′ Q − Q ′ ≠ 0 .

From P ′ Q − Q ′ ≠ 0 , Q ≢ 0 and P ′ ≢ 0 , we know that P ′ and Q are nonzero constants. Thus, P = a z + b 2 and Q = b 3 , where a ( ≠ 0 ) , b 2 and b 3 ( ≠ 0 ) are constants. It follows that f = e a z + b , where b is a constant.

This completes the proof of Theorem 1.8.

Acknowledgments

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

  1. Funding information: This article was supported by the National Natural Science Foundation of China (Grant Nos. 12171127, 12371074).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

[1] W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.Search in Google Scholar

[2] I. Laine, Nevanlinna Theory and Complex Differential Equations, De Gruyter, Berlin, 1993.10.1515/9783110863147Search in Google Scholar

[3] L. Yang, Value Distribution Theory, Springer-Verlag, Berlin, 1993.Search in Google Scholar

[4] C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers Group, Dordrecht, 2003.10.1007/978-94-017-3626-8Search in Google Scholar

[5] J. H. Zhu, A far-reaching form of Hayman’s inequality; fixed points of meromorphic functions and their derivatives, Kexue Tongbao (Chinese) 31 (1986), no. 11, 801–804.10.1360/csb1986-31-11-801Search in Google Scholar

[6] X. H. Hua and C. T. Chuang, On a conjecture of Hayman, Acta Math. Sinica (N.S.) 7 (1991), no. 2, 119–126.10.1007/BF02633943Search in Google Scholar

[7] K. S. Charak and A. Singh, A value distribution result related to Hayman’s alternative, Commun. Korean Math. Soc. 34 (2019), no. 2, 495–506.Search in Google Scholar

[8] W. K. Hayman and J. Miles, On the growth of a meromorphic function and its derivatives, Complex Variables Theory Appl. 12 (1989), no. 1–4, 245–260.10.1080/17476938908814369Search in Google Scholar

[9] C. C. Yang, On the value distribution of a transcendental meromorphic function and its derivatives, Indian J. Pure Appl. Math. 35 (2004), no. 8, 1027–1031.Search in Google Scholar

[10] J. K. Langley, Proof of a conjecture of Hayman concerning f and f″, J. Lond. Math. Soc. (2) 48 (1993), no. 3, 500–514.10.1112/jlms/s2-48.3.500Search in Google Scholar

[11] J. K. Langley, Zeros of derivatives of meromorphic functions, Comput. Methods Funct. Theory 10 (2010), no. 2, 421–439.10.1007/BF03321774Search in Google Scholar
Received: 2023-12-21
Revised: 2024-10-21
Accepted: 2024-12-06
Published Online: 2025-02-04

© 2025 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 Contemporary Developments in Graphs Defined on Algebraic Structures
  2. Forbidden subgraphs of TI-power graphs of finite groups
  3. Finite group with some c#-normal and S-quasinormally embedded subgroups
  4. Classifying cubic symmetric graphs of order 88p and 88p 2
  5. Simplicial complexes defined on groups
  6. Two-sided zero-divisor graphs of orientation-preserving and order-decreasing transformation semigroups
  7. Further results on permanents of Laplacian matrices of trees
  8. Research Articles
  9. Dynamics of particulate emissions in the presence of autonomous vehicles
  10. The regularity of solutions to the Lp Gauss image problem
  11. Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits
  12. The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equation
  13. Some results on value distribution concerning Hayman's alternative
  14. 𝕮-inverse of graphs and mixed graphs
  15. A note on the global existence and boundedness of an N-dimensional parabolic-elliptic predator-prey system with indirect pursuit-evasion interaction
  16. On a question of permutation groups acting on the power set
  17. Chebyshev polynomials of the first kind and the univariate Lommel function: Integral representations
  18. Blow-up of solutions for Euler-Bernoulli equation with nonlinear time delay
  19. Spectrum boundary domination of semiregularities in Banach algebras
  20. Statistical inference and data analysis of the record-based transmuted Burr X model
  21. A modified predictor–corrector scheme with graded mesh for numerical solutions of nonlinear Ψ-caputo fractional-order systems
  22. Dynamical properties of two-diffusion SIR epidemic model with Markovian switching
  23. Classes of modules closed under projective covers
  24. On the dimension of the algebraic sum of subspaces
  25. Periodic or homoclinic orbit bifurcated from a heteroclinic loop for high-dimensional systems
  26. On tangent bundles of Walker four-manifolds
  27. Regularity of weak solutions to the 3D stationary tropical climate model
  28. A new result for entire functions and their shifts with two shared values
  29. Freely quasiconformal and locally weakly quasisymmetric mappings in metric spaces
  30. On the spectral radius and energy of the degree distance matrix of a connected graph
  31. Solving the quartic by conics
  32. A topology related to implication and upsets on a bounded BCK-algebra
  33. On a subclass of multivalent functions defined by generalized multiplier transformation
  34. Local minimizers for the NLS equation with localized nonlinearity on noncompact metric graphs
  35. Approximate multi-Cauchy mappings on certain groupoids
  36. Multiple solutions for a class of fourth-order elliptic equations with critical growth
  37. A note on weighted measure-theoretic pressure
  38. Majorization-type inequalities for (m, M, ψ)-convex functions with applications
  39. Recurrence for probabilistic extension of Dowling polynomials
  40. A generalized fixed-point theorem for set-valued mappings in b-metric spaces
  41. Unraveling chaos: A topological analysis of simplicial homology groups and their foldings
  42. Global existence and blow-up of solutions to pseudo-parabolic equation for Baouendi-Grushin operator
  43. A characterization of the translational hull of a weakly type B semigroup with E-properties
  44. Some new bounds on resolvent energy of a graph
  45. Carmichael numbers composed of Piatetski-Shapiro primes in Beatty sequences
  46. The number of rational points of some classes of algebraic varieties over finite fields
  47. Singular direction of meromorphic functions with finite logarithmic order
  48. Pullback attractors for a class of second-order delay evolution equations with dispersive and dissipative terms on unbounded domain
  49. Eigenfunctions on an infinite Schrödinger network
  50. Boundedness of fractional sublinear operators on weighted grand Herz-Morrey spaces with variable exponents
  51. On SI2-convergence in T0-spaces
  52. Bubbles clustered inside for almost-critical problems
  53. Classification and irreducibility of a class of integer polynomials
  54. Existence and multiplicity of positive solutions for multiparameter periodic systems
  55. Averaging method in optimal control problems for integro-differential equations
  56. On superstability of derivations in Banach algebras
  57. Investigating the modified UO-iteration process in Banach spaces by a digraph
  58. The evaluation of a definite integral by the method of brackets illustrating its flexibility
  59. Existence of positive periodic solutions for evolution equations with delay in ordered Banach spaces
  60. Tilings, sub-tilings, and spectral sets on p-adic space
  61. The higher mapping cone axiom
  62. Continuity and essential norm of operators defined by infinite tridiagonal matrices in weighted Orlicz and l∞ spaces
  63. A family of commuting contraction semigroups on l 1 ( N ) and l ∞ ( N )
  64. Pullback attractor of the 2D non-autonomous magneto-micropolar fluid equations
  65. Maximal function and generalized fractional integral operators on the weighted Orlicz-Lorentz-Morrey spaces
  66. On a nonlinear boundary value problems with impulse action
  67. Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth
  68. Decompositions of the extended Selberg class functions
  69. Subharmonic functions and associated measures in ℝn
  70. Some new Fejér type inequalities for (h, g; α - m)-convex functions
  71. The robust isolated calmness of spectral norm regularized convex matrix optimization problems
  72. Multiple positive solutions to a p-Kirchhoff equation with logarithmic terms and concave terms
  73. Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals
  74. Green's graphs of a semigroup
  75. Some new Hermite-Hadamard type inequalities for product of strongly h-convex functions on ellipsoids and balls
  76. Infinitely many solutions for a class of Kirchhoff-type equations
  77. On an uncertainty principle for small index subgroups of finite fields
  78. On a generalization of I-regularity
  79. Spin(8, ℂ)-Higgs bundles fixed points through spectral data
  80. Coloring the vertices of a graph with mutual-visibility property
  81. Embedding of lattices and K3-covers of an enriques surface
  82. Algorithm and linear convergence of the H-spectral radius of weakly irreducible quasi-positive tensors
  83. q-Stirling sequence spaces associated with q-Bell numbers
  84. Multiple G-Stratonovich integral in G-expectation space
https://doi.org/10.1515/math-2024-0117
Keywords for this article
meromorphic functions; value distribution; differential polynomials; Hayman’s alternative
Creative Commons
BY 4.0

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graphs Defined on Algebraic Structures
  2. Forbidden subgraphs of TI-power graphs of finite groups
  3. Finite group with some c#-normal and S-quasinormally embedded subgroups
  4. Classifying cubic symmetric graphs of order 88p and 88p 2
  5. Simplicial complexes defined on groups
  6. Two-sided zero-divisor graphs of orientation-preserving and order-decreasing transformation semigroups
  7. Further results on permanents of Laplacian matrices of trees
  8. Research Articles
  9. Dynamics of particulate emissions in the presence of autonomous vehicles
  10. The regularity of solutions to the Lp Gauss image problem
  11. Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits
  12. The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equation
  13. Some results on value distribution concerning Hayman's alternative
  14. 𝕮-inverse of graphs and mixed graphs
  15. A note on the global existence and boundedness of an N-dimensional parabolic-elliptic predator-prey system with indirect pursuit-evasion interaction
  16. On a question of permutation groups acting on the power set
  17. Chebyshev polynomials of the first kind and the univariate Lommel function: Integral representations
  18. Blow-up of solutions for Euler-Bernoulli equation with nonlinear time delay
  19. Spectrum boundary domination of semiregularities in Banach algebras
  20. Statistical inference and data analysis of the record-based transmuted Burr X model
  21. A modified predictor–corrector scheme with graded mesh for numerical solutions of nonlinear Ψ-caputo fractional-order systems
  22. Dynamical properties of two-diffusion SIR epidemic model with Markovian switching
  23. Classes of modules closed under projective covers
  24. On the dimension of the algebraic sum of subspaces
  25. Periodic or homoclinic orbit bifurcated from a heteroclinic loop for high-dimensional systems
  26. On tangent bundles of Walker four-manifolds
  27. Regularity of weak solutions to the 3D stationary tropical climate model
  28. A new result for entire functions and their shifts with two shared values
  29. Freely quasiconformal and locally weakly quasisymmetric mappings in metric spaces
  30. On the spectral radius and energy of the degree distance matrix of a connected graph
  31. Solving the quartic by conics
  32. A topology related to implication and upsets on a bounded BCK-algebra
  33. On a subclass of multivalent functions defined by generalized multiplier transformation
  34. Local minimizers for the NLS equation with localized nonlinearity on noncompact metric graphs
  35. Approximate multi-Cauchy mappings on certain groupoids
  36. Multiple solutions for a class of fourth-order elliptic equations with critical growth
  37. A note on weighted measure-theoretic pressure
  38. Majorization-type inequalities for (m, M, ψ)-convex functions with applications
  39. Recurrence for probabilistic extension of Dowling polynomials
  40. A generalized fixed-point theorem for set-valued mappings in b-metric spaces
  41. Unraveling chaos: A topological analysis of simplicial homology groups and their foldings
  42. Global existence and blow-up of solutions to pseudo-parabolic equation for Baouendi-Grushin operator
  43. A characterization of the translational hull of a weakly type B semigroup with E-properties
  44. Some new bounds on resolvent energy of a graph
  45. Carmichael numbers composed of Piatetski-Shapiro primes in Beatty sequences
  46. The number of rational points of some classes of algebraic varieties over finite fields
  47. Singular direction of meromorphic functions with finite logarithmic order
  48. Pullback attractors for a class of second-order delay evolution equations with dispersive and dissipative terms on unbounded domain
  49. Eigenfunctions on an infinite Schrödinger network
  50. Boundedness of fractional sublinear operators on weighted grand Herz-Morrey spaces with variable exponents
  51. On SI2-convergence in T0-spaces
  52. Bubbles clustered inside for almost-critical problems
  53. Classification and irreducibility of a class of integer polynomials
  54. Existence and multiplicity of positive solutions for multiparameter periodic systems
  55. Averaging method in optimal control problems for integro-differential equations
  56. On superstability of derivations in Banach algebras
  57. Investigating the modified UO-iteration process in Banach spaces by a digraph
  58. The evaluation of a definite integral by the method of brackets illustrating its flexibility
  59. Existence of positive periodic solutions for evolution equations with delay in ordered Banach spaces
  60. Tilings, sub-tilings, and spectral sets on p-adic space
  61. The higher mapping cone axiom
  62. Continuity and essential norm of operators defined by infinite tridiagonal matrices in weighted Orlicz and l∞ spaces
  63. A family of commuting contraction semigroups on l 1 ( N ) and l ∞ ( N )
  64. Pullback attractor of the 2D non-autonomous magneto-micropolar fluid equations
  65. Maximal function and generalized fractional integral operators on the weighted Orlicz-Lorentz-Morrey spaces
  66. On a nonlinear boundary value problems with impulse action
  67. Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth
  68. Decompositions of the extended Selberg class functions
  69. Subharmonic functions and associated measures in ℝn
  70. Some new Fejér type inequalities for (h, g; α - m)-convex functions
  71. The robust isolated calmness of spectral norm regularized convex matrix optimization problems
  72. Multiple positive solutions to a p-Kirchhoff equation with logarithmic terms and concave terms
  73. Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals
  74. Green's graphs of a semigroup
  75. Some new Hermite-Hadamard type inequalities for product of strongly h-convex functions on ellipsoids and balls
  76. Infinitely many solutions for a class of Kirchhoff-type equations
  77. On an uncertainty principle for small index subgroups of finite fields
  78. On a generalization of I-regularity
  79. Spin(8, ℂ)-Higgs bundles fixed points through spectral data
  80. Coloring the vertices of a graph with mutual-visibility property
  81. Embedding of lattices and K3-covers of an enriques surface
  82. Algorithm and linear convergence of the H-spectral radius of weakly irreducible quasi-positive tensors
  83. q-Stirling sequence spaces associated with q-Bell numbers
  84. Multiple G-Stratonovich integral in G-expectation space
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 5.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2024-0117/html
Scroll to top button