Singular direction of meromorphic functions with finite logarithmic order

Theorem A

[21] Let f ( z ) be a meromorphic function in the complex plane C with finite logarithmic order ρ , if f ( z ) satisfies the growth condition

then there exists a direction arg z = θ 0 ( 0 ≤ θ 0 < 2 π ) , such that for every small positive number ε and every a ∈ C ˆ , the equation

holds with at most two possible exceptional values of a, where n ( r , θ 0 , ε , f = a ) denotes the number of zeros of f ( z ) − a in the angular region { z : ∣ arg z − θ 0 ∣ ≤ ε , ∣ z ∣ < r } .

Theorem B

[22] Let f ( z ) be a meromorphic function in the complex plane C with finite logarithmic order ρ , then there exists a direction arg z = θ 0 ( 0 ≤ θ 0 < 2 π ) , such that for every small positive number ε and every a ∈ C ˆ , the equation

holds with at most two possible exceptional values of a.

Theorem 1.1

Let f ( z ) be a meromorphic function in the complex plane C with finite logarithmic order, that for every sufficiently large R and k > 1 , satisfies the following inequality in the annulus r < ∣ z ∣ < R :

Then, there must exist a point z j in the annulus r < ∣ z ∣ < R such that f ( z ) takes every complex number at least n = C T ( R , f ) q 2 log q log R r log log R r times in ∣ z − z j ∣ < 4 π log q − 1 ∣ z j ∣ , where C is a constant and q is sufficiently large positive integer, except possibly for those numbers contained in two spherical disks each with radius e − n .

Theorem 1.2

Let f ( z ) be a meromorphic function in the complex plane C with finite logarithmic order ρ . If f ( z ) satisfies the growth condition (1.1), then there exists a sequence of disks

such that f ( z ) takes every complex number at least ( log ∣ z j ∣ ) ρ − 1 − δ j times in Γ j , where lim j → ∞ δ j = 0 , except possibly for those numbers contained in two spherical disks each with radius e − ( log ∣ z j ∣ ) ρ − 1 − δ j .

Theorem 1.3

Let f ( z ) be a meromorphic function in the angular domain ∣ arg z − θ 0 ∣ < η , and there are three distinct values a ν ( ν = 1, 2, 3) and a positive number σ , such that

converges, where ν = 1, 2, 3, and r j ( θ 0 , η , f = a ν ) ( j = 1 , 2 , … ) denote the moduli of the zeros of f ( z ) − a ν in the domain { z : ∣ arg z − θ 0 ∣ < η , ∣ z ∣ > 1 } , arranged by nondecreasing order and counted with their multiplicities. Then, ∑ j ( log r j ( θ 0 , η − ε , f = a ) ) − σ is convergent, for any positive number ε < η and all a ∈ C ˆ , possibly except at most for a set whose line measure is zero.

Theorem 1.4

Let f ( z ) be a meromorphic function in the complex plane C with finite logarithmic order λ + 1 , if f ( z ) has no Borel direction of finite logarithmic order λ in the angular domain θ 1 < arg z < θ 2 , then for any small positive number α , there exist three distinct values a ν ( ν = 1, 2, 3) and a positive number τ ( < λ ) , such that

where n ( r , θ 1 + α , θ 2 − α , f = a ν ) denotes the number of zeros of f ( z ) − a in the domain { z : θ 1 + α ≤ arg z ≤ θ 2 − α , 1 < ∣ z ∣ < r } .

Theorem 1.5

Let f ( z ) be a meromorphic function in the complex plane C with finite logarithmic order λ + 1 > 2 , if

is a Borel direction of finite logarithmic order λ for f ( z ) , then there exists a sequence of disks

such that f ( z ) takes every complex number at least ( log ∣ z j ∣ ) λ − δ j times in Γ j , where lim j → ∞ δ j = 0 , except possibly for those numbers contained in two spherical disks each with radius j − 3 .

Lemma 2.1

[4] Assume that f ( z ) is a meromorphic function in the complex plane C , and D is a bounded region. Divide D into p subregions D j ( j = 1 , 2 , … , p ) . For each D j , make a disk K j ⊃ D j , and then make a concentric disk K j ′ of K j , such that the radius is twice as large as K j . If the set of complex number a such that n ( D , f = a ) ≥ N cannot be covered by a collection of spherical disks with radius total equal to 1 2 on the Riemann sphere, then there must exist a circle K j ′ with a constant C such that f ( z ) takes every complex number at least C N p times in K j ′ , except possibly for those numbers contained in two spherical disks each with radius e − C N p .

Lemma 2.2

[4] Assume that f ( z ) is a meromorphic function in the disk ∣ z ∣ < R , and let

where a 1 , a 2 , and a 3 are three distinct complex numbers, with their spherical distances larger than a positive number d. Then, there exists a point z 0 , with ∣ z 0 ∣ < R , such that for every r ∈ ( 0 , R ) , and any complex number a,

where C is a positive constant, and ∣ f ( z 0 ) , a ∣ denotes the spherical distance between f ( z 0 ) and a.

Lemma 2.3

[4] Let a μ ( μ = 1 , 2 , … , n ) be n complex numbers and h a positive number. Then, the points that satisfy the inequality

can be covered by a collection of spherical disks whose number does not exceed n and the sum of whose radii does not exceed 2 h .

Lemma 2.4

Assume that f ( z ) is a meromorphic function in the complex plane C , and let r j ( j = 1 , 2 , … ) be the moduli of the poles of f ( z ) in the domain { z : ∣ z ∣ > 1 } , with r j ( a ) ≤ r j + 1 ( a ) . For any r 0 > 1 , if σ > 0 , then the series ∑ ( log r j ) − σ and ∫ r 0 ∞ n ( t , f ) ( log t ) σ + 1 t d t are either simultaneously convergent or simultaneously divergent.

Proof

For any R > r 0 with r 0 > 1 , let z j 0 , … , z J be the poles of f within the annulus { z : r 0 < ∣ z ∣ < R } , and denote r j 0 = ∣ z j 0 ∣ , … , r J = ∣ z J ∣ , arranged by nondecreasing order and counted with their multiplicities. It follows from the equation

if ∑ ( log r j ) − σ converges, then ∫ r 0 ∞ n ( t , f ) ( log t ) σ + 1 t d t also converges.

Conversely, if the series ∫ r 0 ∞ n ( t , f ) ( log t ) σ + 1 t d t converges, combine

with (2.1), ∑ ( log r j ) − σ also converging.

So ∑ ( log r j ) − σ and ∫ r 0 ∞ n ( t , f ) ( log t ) σ + 1 t d t are either simultaneously convergent or simultaneously divergent.□

Lemma 2.5

Assume that f ( z ) is a meromorphic function in the complex plane C , 0 ≤ θ 0 < 2 π , η > 0 , a ∈ C . Let r j ( θ 0 , η , f = a ) be the moduli of the a-points of f ( z ) in the domain { z : ∣ arg z − θ 0 ∣ < η , ∣ z ∣ > 1 } , with r j ( a ) ≤ r j + 1 ( a ) . For any r 0 > 1 , if σ > 0 , then the series ∑ ( log r j ( θ 0 , η , f = a ) ) − σ and ∫ r 0 ∞ n ( t , θ 0 , η , f = a ) ( log t ) σ + 1 t d t are either simultaneously convergent or simultaneously divergent.

Proof of Theorem 1.1

Assume that f ( z ) is a meromorphic function in the complex plane C , by the definitions of n ( r , f = a ) and N ( r , f = a ) , we have

and

From the second fundamental theorem of Nevanlinna, for suitably large R and every complex number a , we have

except possibly for those numbers contained in a sequence of disks with total spherical radius 1 2 . Combining (1.2) with (3.3), we have

Taking a sufficiently large positive integer q , the annulus r < ∣ z ∣ < R is divided as follows:

Make q rays from the origin, the angle between two consecutive rays is 2 π q , and then make 1 + s [ log q ] circles

where s = log log R r log ( 1 + 2 π q ) + 1 .

Thus, the annulus r < ∣ z ∣ < R is divided into p curved quadrilaterals D j , where

Let z j ( j = 1 , … , p ) be the center of D j , then D j ⊂ K j = { z : ∣ z − z j ∣ < 2 π log q − 1 ∣ z j ∣ } . The division of the annulus is shown in Figure 1.

By Lemma 2.1, then there must exist a concentric circle K j ′ = { z : ∣ z − z j ∣ < 4 π log q − 1 ∣ z j ∣ } with a constant C , such that f ( z ) takes every complex number at least n = C T ( R , f ) q 2 log q log R r log log R r times in K j ′ , except possibly for those numbers contained in two spherical disks each with radius e − n .□

Proof of Theorem 1.2

Since f ( z ) is a meromorphic function with finite logarithmic order ρ and (1.1) holds, there exists a sequence { r k } that tends to infinity such that

Taking a suitably large r k 1 , such that

and

By Theorem 1.1, there exists a point ∣ z 1 ∣ in the annulus 1 < ∣ z ∣ < r k 1 such that f ( z ) takes every complex number at least n 1 = C T ( r k 1 , f ) log r k 1 ( log log r k 1 ) 3 log log log r k 1 times in Γ 1 : ∣ z − z 1 ∣ < 4 π log log log r k 1 − 1 ∣ z 1 ∣ , except possibly for those numbers contained in two spherical disks each with radius e − n 1 , where C is a constant.

Take r k 1 ′ such that r k 1 ′ > 2 r k 1 . Then, take r k 2 such that

and

By Theorem 1.1, there exists a point ∣ z 2 ∣ in the annulus r k 1 ′ < ∣ z ∣ < r k 2 such that f ( z ) takes every complex number at least n 2 = C T ( r , f ) log r k 2 ( log log r k 2 ) 3 log log log r k 2 times in Γ 2 : ∣ z − z 2 ∣ < 4 π log log log r k 2 − 1 ∣ z 2 ∣ , except possibly for those numbers contained in two spherical disks each with radius e − n 2 , where C is a constant.

By induction, we can obtain a sequence of disks

such that f ( z ) takes every complex number at least

times in Γ j , except possibly for those numbers contained in two spherical disks each with radius e − ( log ∣ z j ∣ ) ρ − 1 − δ j , where C is a constant and lim j → ∞ δ j = 0 .□

Proof of Theorem 1.3

Split the angular domain ∣ arg z − θ 0 ∣ < η − ε with rays from the origin so that the angle of each small angular domain does not exceed ε 4 , the total number of these small angular domains is J ≥ 4 ( η − ε ) ε 4 + 1 . Then, take a suitably large positive number r 0 > 1 , take the origin as the center, and make circles ∣ z ∣ = r 0 , r 0 exp { 1 + ε 4 } , r 0 exp { ( 1 + ε 4 ) 2 } , … . such that ( ∣ z ∣ > r 0 ) ∩ ( ∣ arg z − θ 0 ∣ < η − ε ) is divided into some small curved quadrilateral Ω j l ( j = 1 , 2 , … , J ; l = 1 , 2 , … , L ). For each Ω j l , there exists a small disk Γ j l and a disk Γ j l ′ of radius two times its radius, such that Ω j l ⊂ Γ j l ⊂ Γ j l ′ ⊂ ( ∣ arg z − θ 0 ∣ < η ) . It is easy to see that for each Γ j 0 l 0 ′ , there is a fixed upper bound for the number of other Γ j l ′ intersecting it. At the same time, combining (1.3) with Lemma 2.5, we obtain

By Lemmas 2.2 and 2.3, for any a ∈ C ˆ , the equation

holds in each Γ j l ′ , except possibly for those numbers contained in the disk D j l with spherical radius exp − σ 2 log ∣ r 0 ∣ + ( 1 + ε 4 ) l .

Let D l = ∪ j = 1 J D j l , D = ∩ k = 1 ∞ ( ∪ l = k ∞ D l ) , and we see that mes D = lim k → ∞ mes ( ∪ l = k ∞ D l ) = 0 .

For any complex a ∉ D , there exists l 0 , such that a ∉ ∪ l = l 0 ∞ D l . So when l ≥ l 0 , (4.2) holds for j = 1 , 2 , … , J .

Thus,

which implies

By (4.1), for any complex a ∉ D , we have

Furthermore, by Lemma 2.5, the theorem is proved.□

Proof of Theorem 1.4

According to the conditions of the theorem, for any φ ∈ [ θ 1 + α , θ 2 − α ] , arg z = φ is not a Borel direction of finite logarithmic order for f ( z ) . Nevertheless, there must exist three distinct values β j ( j = 1, 2, 3) and positive numbers ε ( φ ) , τ ( φ ) ( < λ ) , such that

It follows that, for τ 1 ( φ ) ( τ ( φ ) < τ 1 ( φ ) < λ ) and r 0 > 1 , we have

By Lemma 2.5, the series ∑ ( log r ( φ , ε ( φ ) , f = β j ) ) − τ 1 ( φ ) is convergent. As a result of applying Theorem 1.3, for any complex number a , the series ∑ ( log r ( φ , ε ( φ ) 2 , f = a ) ) − τ 1 ( φ ) is convergent possibly except for at most a set D ( φ ) whose line measure is zero.

We note that { ( φ − ε ( φ ) 2 , φ + ε ( φ ) 2 ) : θ 1 + α ≤ φ ≤ θ 2 − α } forms an open covering of the closed interval [ θ 1 + α , θ 2 − α ] . Therefore, there exists a finite set of intervals ( φ l − ε l 2 , φ l + ε l 2 ) ( l = 1 , 2 , … , L ) that are an open covering of [ θ 1 + α , θ 2 − α ] .

Corresponding to each φ l , its exceptional zero measure set is D l = D ( φ l ) . Let D = ∪ l = 1 L D l , then it is still the zero measure set. If τ 1 = max 1 ≤ l ≤ L { τ 1 ( φ l ) } , then 0 < τ 1 < λ .

Hence, for every complex number a ( ∉ D ) , we obtain

So

By Lemma 2.5, we obtain

In particular, this holds for any three distinct complex numbers a ν ( ν = 1, 2, 3) that do not belong to D , and positive numbers τ ( τ 1 < τ < λ ) . Hence, we obtain

Proof of Theorem 1.5

Choose a small angular region with B as its bisector, which is defined by

Using the arcs of circumferences ∣ z ∣ = 2 j ( j = 1 , 2 , … ) , divide Ω into a sequence of small quadrangles

Subsequently, each Ω j is further divided by s − 1 arcs of ∣ z ∣ = 2 j ( 1 + η ) l , ( l = 1 , 2 , … , s − 1 ) , where s is a positive integer determined by 2 j ( 1 + η ) s − 1 < 2 j + 1 ≤ 2 j ( 1 + η ) s . This results in s smaller quadrangles, denoted as Ω j l ( l = 1 , 2 , … , s ) . Choose concentric disks Γ j l and Γ j l ′ with radius C ⋅ 2 j + 1 η and 2 C ⋅ 2 j + 1 η , respectively, such that

Define n j l as follows:

where the minimum is taken over all the triples of complex numbers, provided that the mutual spherical distances among these three complex numbers are at least j − 3 . By Lemma 2.2, f ( z ) takes every complex number a at most C { n j l + [ log j ] + 1 } times in Ω j l , except for those complex numbers contained in a disk with spherical radius j − 3 .

For each fixed value of j , there exist s exceptional spherical disks. As j varies, the aggregate radius of these exceptional spherical disks is given by the series:

By selecting a sufficiently large value for j 0 , we can ensure that

Since B is a Borel direction of logarithmic order λ of f ( z ) , by Lemma 2.5, the series