The main purpose of this paper is to investigate the growth of several entire functions represented by double Dirichlet series of finite logarithmic order, h-order. Besides, we also study some properties on the maximum modulus of double Dirichlet series and its partial derivative. Our results are extension and improvement of previous results given by Huo and Liang.
National Natural Science Foundation of China11561033Natural Science Foundation of Jiangxi Province20181BAB201001Foundation of Education Department of JiangxiGJJ170759GJJ1707881. Introduction and Basic Notes
For Dirichlet series(1)fs=∑n=1∞aneλns,s=σ+it,where(2)0≤λ1<λ2<⋯<λn<⋯,λn→+∞asn→+∞;s=σ+it (σ,t are real variables); an are nonzero complex numbers.
It is an interesting topic to study some properties of Dirichlet series in the fields of complex analysis;particularly, considerable attention has been paid to the analytic function and entire funtcions represented by Dirichlet series in the half-plane and whole plane, and a number of interesting and important results can be found in [1]. For example, J. R. Yu, G. Srivastava, P. V. Filevich, Z. S. Gao, D. C. Sun, etc. studied the growth and value distribution of Dirichlet series and random Dirichlet series (see [2–17]); Y. Y. Kong, G. T. Deng investigated the growth of Dirichlet-Hadamard product (see [18, 19]); A. R. Reddy, M. N. Sheremeta, C. F. Yi, and H. Y. Xu studied the approximation of Dirichlet series (see [20–24]), and so on.
In 1962, J. R. Yu [25] had made some pioneering research for the growth of double Dirichlet series as follows:(3)fs1,s2=∑m,n=0+∞am,nexpλms1+μns2,where s1=σ1+it1,s2=σ2+it2, am,n∈C, and(4)0=λ0<λ1<⋯<λm<⋯↑+∞,0=μ0<μ1<⋯<μn<⋯↑+∞,n→+∞.
However, there were few results about double Dirichlet series because the research involves complex two-dimensional space. In 2009, J. Liu and Z. S. Gao [26] discussed the problem on θ-order of entire function represented by the double Dirichlet series in the double horizontal line; recently, G. N. Gao [27] further studied the problem about θ-order of double Dirichlet series by using the Knopp-Kojima method. In this paper, we further investigate the growth of entire functions represented by double Dirichlet series, such as the logarithmic order, h-order, and some properties of the maximum modulus of double Dirichlet series and its partial derivatives.
If double Dirichlet series satisfies(5)limsupm+n→+∞logm+nλm+μn=E<+∞,limsupm+n→+∞logam,nλm+μn=-∞,then we call that f(s1,s2) is analytic in the double whole plane, i.e., the entire Dirichlet series of double whole planes. Let D be the set of all entire function represented by double Dirichlet series (3) satisfying (4)-(5). Let(6)Mσ1,σ2,f=sup-∞<t1,t2<+∞fσ1+it1,σ2+it2,Res1≤σ1,Res2≤σ2,be the maximum modulus of f(s1,s2) in Res1≤σ1,Res2≤σ2. From the definition of the maximum modulus, if f(s1,s2) is nonconstant with respect to s1,s2, we have(7)Mσ1′,σ2,f>Mσ1,σ2,f,σ1′>σ1;Mσ1,σ2′>Mσ1,σ2,σ2′>σ2;and(8)Mσ1′,σ2′,f>Mσ1,σ2,f,σ1′>σ1,σ2′>σ2.
To state our results, we can introduce the following definitions.
Definition 1.
Suppose f(s1,s2)∈D; let L be the set of f(s1,s2)(∈D) with finite logarithmic order and f(s1,s2) satisfy the following conditions:
For any fixed value of σ2>0, there exists σ10=σ10(K1,β1,σ2) such that(9)Mσ1,σ2,f<expK1σ1β1,forσ1≥σ10,
For any fixed value of σ1>0, there exists σ20=σ20(K2,β2,σ1) such that(10)Mσ1,σ2,f<expK2σ2β2,forσ2≥σ20,
where K1>0,β1>0;K2>0,β2>0 are constants, so there exists σ=σ(K1,β1,K2,β2) such that(11)Mσ1,σ2,f<expK1σ1β1+K2σ2β2,forσ1,σ2≥σ.
Definition 2.
Suppose that f(s1,s2)∈D; for any small ε>0 and fixed value σ2>0, there exists σ(1)=σ(1)(ε,σ2) such that(12)Mσ1,σ2,f<expσ1ρ1L+ε,forσ1≥σ1,and there exists at least a real value σ20(ε) and sufficiently large number σ1i such that(13)Mσ1i,σ20ε,f>expσ1iρ1L-ε,then we say that f(s1,s2) has partial logarithmic order ρ1L with respect to s1; similarly, for any small ε>0 and fixed value σ1>0, there exists σ(2)=σ(2)(ε,σ2) such that(14)Mσ1,σ2,f<expσ2ρ2L+ε,forσ2≥σ2,and there exists at least a real value σ10(ε) and sufficiently large number σ2j such that(15)Mσ2j,σ10ε,f>expσ2jρ2L-ε,then we say that f(s1,s2) has partial logarithmic order ρ2L with respect to s2.
Remark 3.
If f(s1,s2) has partial logarithmic order ρ1L with respect to s1, that is,(16)limsupσ2→+∞limsupσ1→+∞loglogMσ1,σ2,flogσ1=ρ1L;and if f(s1,s2) has partial logarithmic order ρ2 with respect to s2, that is,(17)limsupσ1→+∞limsupσ2→+∞loglogMσ1,σ2,flogσ2=ρ2L.
Definition 4.
Let f(s1,s2)∈D, 0<ρ1L,ρ2L<∞, and satisfy the following conditions:
f(s1,s2)∈L;
f(s1,s2) has partial logarithmic order ρ1L with respect to s1 and partial logarithmic order ρ2L with respect to s2;
for any small number ε>0, there exists σ=σ(ε) such that(18)Mσ1,σ2,f<expσ1ρ1L+ε+σ2ρ2L+ε,asσ1,σ2>σ,
then we say that f(s1,s2) has finite logarithmic order (ρ1L,ρ2L).
For the logarithmic order of double Dirichlet series f(s1,s2), we have the following.
Theorem 5.
Let f(s1,s2)∈D be of finite logarithmic order (ρ1L,ρ2L), 1<ρ1L,ρ2L<+∞; then(19)limsupσ1,σ2→+∞loglogMσ1,σ2,fρ1Llogσ1+ρ2Llogσ2=1.
To estimate the growth of f(s1,s2) more precisely, we will give the logarithmic type of f(s1,s2) as follows.
Definition 6.
Suppose f(s1,s2)∈D; let LT be the set of f(s1,s2)∈L given by double Dirichlet series of finite logarithmic order (ρ1L,ρ2L) has finite logarithmic type and f(s1,s2) satisfy the following conditions:
For any fixed value of σ2>0, there exists σ10=σ10(K1,σ2) such that(20)Mσ1,σ2,f<expK1ρ1Lσ1,forσ1≥σ10,
For any fixed value of σ1>0, there exists σ20=σ20(K2,σ1) such that(21)Mσ1,σ2,f<expK2ρ2Lσ2,forσ2≥σ20,
where K1>0,K2>0, are constants, so there exists σ=σ(K1,K2,) such that(22)Mσ1,σ2,f<expK1ρ1Lσ1+K2ρ2Lσ2,forσ1,σ2≥σ.
Definition 7.
Suppose that f(s1,s2)∈D is of finite logarithmic (ρ1L,ρ2L), and for any small ε>0 and fixed value σ2>0, there exists σ(1)=σ(1)(ε,σ2) such that(23)Mσ1,σ2,f<expτ1L+ερ1Lσ1,forσ1≥σ1,and there exist at least a real value σ20(ε) and sufficiently large number σ1i such that(24)Mσ1i,σ20ε,f>expτ1L-ερ1Lσ1i,then we say that f(s1,s2) has partial logarithmic type τ1L with respect to s1; similarly, for any small ε>0 and fixed value σ1>0, there exists σ(2)=σ(2)(ε,σ2) such that(25)Mσ1,σ2,f<expτ2L+ερ2Lσ2,forσ2≥σ2,and there exists at least a real value σ10(ε) and sufficiently large number σ2j such that(26)Mσ2j,σ10ε,f>expτ2L-ερ2Lσ2j,then we say that f(s1,s2) have partial logarithmic type τ2L with respect to s2.
Remark 8.
If f(s1,s2) is of finite logarithmic order (ρ1L,ρ2L) having partial logarithmic type τ1L with respect to s1, that is,(27)limsupσ2→+∞limsupσ1→+∞logMσ1,σ2,fρ1Lσ1=τ1L;and if f(s1,s2) has partial logarithmic order ρ2 with respect to s2, that is,(28)limsupσ1→+∞limsupσ2→+∞logMσ1,σ2,fρ2Lσ2=τ2L.
Definition 9.
Let f(s1,s2)∈D be of finite logarithmic order (ρ1L,ρ2L), 0<ρ1L,ρ2L<∞, and satisfy the following conditions:
f(s1,s2)∈LT;
f(s1,s2) has partial logarithmic type τ1L with respect to s1 and partial logarithmic type τ2L with respect to s2;
for any small number ε>0, there exists σ=σ(ε) such that(29)Mσ1,σ2,f<expτ1L+ερ1Lσ1+τ2L+ερ2Lσ2,asσ1,σ2>σ,
then we say that f(s1,s2) has finite logarithmic type (τ1L,τ2L).
For the logarithmic type of double Dirichlet series f(s1,s2), we have the following.
Theorem 10.
Let f(s1,s2)∈D be of finite logarithmic order (ρ1L,ρ2L), 1<ρ1L,ρ2L<+∞, and of finite logarithmic type (τ1L,τ2L); then(30)limsupσ1,σ2→+∞logMσ1,σ2,fτ1Lρ1Lσ1+τ2Lρ2Lσ2=1.
In this paper, we also deal with the growth of double Dirichlet series of finite order and infinite order by using a class of functions to reduce M(σ1,σ2,f) which is better than the previous form. So, we firstly give the definition of h-order of double Dirichlet series as follows, which is an extension of [6, 9].
Let F be the class of all functions h(x) satisfying the following conditions:
h(x) is defined on [a,+∞) and is positive, strictly increasing, and differentiable and tends to +∞ as x→+∞;
h(x)~logx as x→+∞ for p=1, and limx→+∞d(h(x))/d(logpx)=0, p>1,p∈N+, where log0x=x,log1x=logx and logpx=log(logp-1x).
Similar to the above definitions, we give the h-order of double Dirichlet series as follows.
Definition 11.
Suppose f(s1,s2)∈D and h(x)∈F; let Lh be the set of f(s1,s2)(∈D) with finite h-order, and f(s1,s2) satisfies the following conditions:
For any fixed value of σ2>0, there exists σ10=σ10(K1,β1,σ2) such that(31)Mσ1,σ2,f<expγK1σ1β1,forσ1≥σ10,
For any fixed value of σ1>0, there exists σ20=σ20(K2,β2,σ1) such that(32)Mσ1,σ2,f<expγK2σ2β2,forσ2≥σ20,
where K1>0,β1>0;K2>0,β2>0 are constants, so there exists σ=σ(K1,β1,K2,β2) such that(33)Mσ1,σ2,f<expγK1σ1β1+K2σ2β2,forσ1,σ2≥σ,where γ(x) is the inverse function of h(x), especially γ(x)=ex as p=1.
Definition 12.
Suppose that f(s1,s2)∈D; for any small ε>0 and fixed value σ2>0, there exists σ(1)=σ(1)(ε,σ2) such that(34)Mσ1,σ2,f<expγσ1ρ1h+ε,forσ1≥σ1,and there exist at least a real value σ20(ε) and sufficiently large number σ1i such that(35)Mσ1i,σ20ε,f>expγσ1iρ1h-ε,then we say that f(s1,s2) has partial h-order ρ1h with respect to s1; similarly, for any small ε>0 and fixed value σ1>0, there exists σ(2)=σ(2)(ε,σ2) such that(36)Mσ1,σ2,f<expγσ2ρ2h+ε,forσ2≥σ2,and there exist at least a real value σ10(ε) and sufficiently large number σ2j such that(37)Mσ2j,σ10ε,f>expγσ2jρ2h-ε,then we say that f(s1,s2) has partial h-order ρ2h with respect to s2.
Remark 13.
If f(s1,s2) has partial h-order ρ1h with respect to s1, that is,(38)limsupσ2→+∞limsupσ1→+∞hlogMσ1,σ2,fσ1=ρ1h;and if f(s1,s2) has partial h-order ρ2h with respect to s2, that is,(39)limsupσ1→+∞limsupσ2→+∞hlogMσ1,σ2,fσ2=ρ2h.
Definition 14.
Let f(s1,s2)∈D, 0<ρ1h,ρ2h<∞, and satisfy the following conditions:
f(s1,s2)∈Lh;
f(s1,s2) has partial h-order ρ1h with respect to s1 and partial h-order ρ2h with respect to s2;
for any small number ε>0, there exists σ=σ(ε) such that (40)Mσ1,σ2,f<expγσ1ρ1h+ε+σ2ρ2h+ε,asσ1,σ2>σ,
then we say that f(s1,s2) has finite h-order (ρ1h,ρ2h).
For h-order of double Dirichlet series, we have the following.
Theorem 15.
Let f(s1,s2)∈D and be of finite h-order (ρ1h,ρ2h), 0<ρ1h,ρ2h<+∞; then(41)limsupσ1,σ2→+∞hlogMσ1,σ2,fρ1hσ1+ρ2hσ2=1.
In fact, when h(x)=log(x), then h-order is called finite order. Thus, we obtain the following conclusion.
Corollary 16.
Let f(s1,s2)∈D be of finite order (ρ1,ρ2), 0<ρ1,ρ2<∞; then(42)limsupσ1,σ2→+∞loglogMσ1,σ2,fρ1σ1+ρ2σ2=1.
And when h(x)=logpx,p≥2 and p∈N+, then h-order is called the p-order. So, we have the following.
Corollary 17.
Let f(s1,s2)∈D be of finite p-order (ρ1p,ρ2), 0<ρ1p,ρ2p<+∞; then(43)limsupσ1,σ2→+∞logp+1Mσ1,σ2,fρ1pσ1+ρ2pσ2=1.
Similarly, we give the h-type of double Dirichlet series f(s1,s2) as follows.
Definition 18.
Suppose f(s1,s2)∈D and h(x)∈F; let LTh be the set of f(s1,s2)∈Lh given by double Dirichlet series of finite logarithmic order (ρ1L,ρ2L) having finite h-order, and f(s1,s2) satisfies the following conditions:
For any fixed value of σ2>0, there exists σ10=σ10(K1,β1,σ2) such that(44)Mσ1,σ2,f<γK1expρ1hσ1,forσ1≥σ10,
For any fixed value of σ1>0, there exists σ20=σ20(K2,β2,σ1) such that(45)Mσ1,σ2,f<γK2expρ2hσ2,forσ2≥σ20,
where K1>0,β1>0;K2>0,β2>0 are constants, so there exists σ=σ(K1,β1,K2,β2) such that(46)Mσ1,σ2,f<γK1expρ1hσ1+K2expρ2hσ2,forσ1,σ2≥σ.
Definition 19.
Suppose that f(s1,s2)∈D; for any small ε>0 and fixed value σ2>0, there exists σ(1)=σ(1)(ε,σ2) such that(47)Mσ1,σ2,f<γτ1h+εexpρ1hσ1,forσ1≥σ1,and there exist at least a real value σ20(ε) and sufficiently large number σ1i such that(48)Mσ1i,σ20ε,f>expγ(τ1h-εexpρ1hσ1i,then we say that f(s1,s2) has partial h-type ρ1h with respect to s1; similarly, for any small ε>0 and fixed value σ1>0, there exists σ(2)=σ(2)(ε,σ2) such that(49)Mσ1,σ2,f<γτ2h+εexpρ2hσ2,forσ2≥σ2,and there exist at least a real value σ10(ε) and sufficiently large number σ2j such that(50)Mσ2j,σ10ε,f>γτ2h-εexpρ2hσ2i,then we say that f(s1,s2) has partial h-type ρ2h with respect to s2.
Remark 20.
If f(s1,s2) has partial h-type τ1h with respect to s1, that is,(51)limsupσ2→+∞limsupσ1→+∞hMσ1,σ2,fexpρ1hσ1=τ1h;and if f(s1,s2) has partial h-order ρ2h with respect to s2, that is,(52)limsupσ1→+∞limsupσ2→+∞hMσ1,σ2,fexpρ2hσ2=τ2h.
Definition 21.
Let f(s1,s2)∈D be of finite h-order (ρ1h,ρ2h), 0<ρ1h,ρ2h<+∞, and satisfy the following conditions:
f(s1,s2)∈Lh;
f(s1,s2) has partial h-type τ1h with respect to s1 and partial h-type τ2h with respect to s2;
for any small number ε>0, there exists σ=σ(ε) such that (53)Mσ1,σ2,f<γτ1h+εexpρ1hσ1+τ2h+εexpρ2hσ2,asσ1,σ2>σ,
then we say that f(s1,s2) has finite h-type (τ1h,τ2h).
For h-type of double Dirichlet series f(s1,s2), we have the following.
Theorem 22.
Let f(s1,s2)∈D and be of finite h-order (ρ1h,ρ2h), 0<ρ1h,ρ2h<+∞, and of finite h-type (τ1h,τ2h); then(54)limsupσ1,σ2→+∞hMσ1,σ2,fτ1hexpρ1hσ1+τ2hexpρ2hσ2=1.
Particularly, we can get the following corollaries.
Corollary 23.
Let f(s1,s2)∈D and be of finite order (ρ1,ρ2) and of finite type (τ1,τ2); then(55)limsupσ1,σ2→+∞logMσ1,σ2,fτ1expρ1σ1+τ2expρ2σ2=1.
Corollary 24.
Let f(s1,s2)∈D and be of finite p-order (ρ1p,ρ2p), 0<ρ1p,ρ2p<+∞, and of finite p-type (τ1p,τ2p) as p≥2; then(56)limsupσ1,σ2→+∞logpMσ1,σ2,fτ1pexpρ1pσ1+τ2pexpρ2pσ2=1.
Remark 25.
In 2010 and 2015, Liang and Gao [28–30] further investigated the growth and convergence of n-multiple Dirichlet series. We only listed some Liang’s definitions as n=2; Liang defined the order of multiple Dirichlet series f(s1,s2) by(57)ρ=limsupσ1,σ2→+∞loglogMσ1,σ2,flogeσ1+eσ2,and if ρ∈(0,+∞), then the type of multiple Dirichlet series f(s1,s2)(58)T=limsupσ1,σ2→+∞logMσ1,σ2,flogeρσ1+eρσ2.
Obviously, our definitions about the order and type of double Dirichlet series are more general than Liang’s.
The other purpose of this paper is to investigate some relation between the partial derivatives fsj(s1,s2),(j=1,2) and growth of f(s1,s2). In order to state our results, we first give some notations as follows. Let(59)fsjs1,s2=∂∂sjfs1,s2,and(60)Mjσ1,σ2,f=sup-∞<t1,t2<+∞fsjσ1+it1,σ2+it2,for j=1,2.
Theorem 26.
Let f(s1,s2)∈D be of finite logarithmic order (ρ1L,ρ2L), 1<ρ1L,ρ2L<+∞; then(61)limsupσ2→∞limsupσ1→+∞logM1σ1,σ2,f-logMσ1,σ2,flogσ1=ρ1L-1,and(62)limsupσ1→+∞limsupσ2→+∞logM2σ1,σ2,f-logMσ1,σ2,flogσ2=ρ2L-1.
Theorem 27.
Let f(s1,s2)∈D be of finite logarithmic order (ρ1L,ρ2L), 1<ρ1L,ρ2L<+∞; then(63)limsupσ1,σ2→+∞logM1σ1,σ2,f+M2σ1,σ2,f-logMσ1,σ2,fρ1L-1logσ1+ρ2L-1logσ2=1.
Theorem 28.
Let f(s1,s2)∈D be of finite h-order (ρ1h,ρ2h), 0<ρ1h,ρ2h<+∞; then(64)limsupσ2→+∞limsupσ1→+∞hM1σ1,σ2,f/Mσ1,σ2,fσ1=ρ1h,and(65)limsupσ1→+∞limsupσ2→+∞hM2σ1,σ2,f/Mσ1,σ2,fσ2=ρ2h.
Theorem 29.
Let f(s1,s2)∈D be of finite h-order (ρ1h,ρ2h), 0<ρ1h,ρ2h<+∞; then(66)limsupσ1,σ2→+∞hM1σ1,σ2,f+M2σ1,σ2,f/Mσ1,σ2,fρ1hσ1+ρ2hσ2=1.
For finite order and finite p-order, some corollaries are obtained below.
Corollary 30.
Let f(s1,s2)∈D be of finite order (ρ1,ρ2), 0<ρ1,ρ2<+∞; then(67)limsupσ2→+∞limsupσ1→+∞logM1σ1,σ2,f-logMσ1,σ2,fσ1=ρ1,and(68)limsupσ2→+∞limsupσ1→+∞logM2σ1,σ2,f-logMσ1,σ2,fσ2=ρ2.
Corollary 31.
Let f(s1,s2)∈D be of finite order (ρ1,ρ2), 0<ρ1,ρ2<+∞; then(69)limsupσ1,σ2→+∞logM1σ1,σ2,f+M2σ1,σ2,f-logMσ1,σ2,fρ1σ1+ρ2σ2=1.
Corollary 32.
Let f(s1,s2)∈D be of finite p-order (ρ1p,ρ2p), 0<ρ1p,ρ2p<+∞, p≥2; then(70)limsupσ2→+∞limsupσ1→+∞logpM1σ1,σ2,f/Mσ1,σ2,fσ1=ρ1p,and(71)limsupσ2→+∞limsupσ1→+∞logpM2σ1,σ2,f/Mσ1,σ2,fσ2=ρ2p.
Corollary 33.
Let f(s1,s2)∈D be of finite p-order (ρ1p,ρ2p), 0<ρ1p,ρ2p<+∞,p≥2; then(72)limsupσ1,σ2→+∞logpM1σ1,σ2,f+M2σ1,σ2,f/Mσ1,σ2,fρ1pσ1+ρ2pσ2=1.
2. Proofs of Theorems 5–222.1. The Proof of Theorem 5
From (18), let μ1>ρ1L,μ2>ρ2L; it follows(73)limsupσ1,σ2→+∞loglogMσ1,σ2μ1logσ1+μ2logσ2≤1.When σ1>0,σ2>0, let μ=maxμ1,μ2,ρL=minρ1L,ρ2L; then (74)loglogMσ1,σ2,fρ1Llogσ1+ρ2Llogσ2≤loglogMσ1,σ2,fρLlogσ1+logσ2=loglogMσ1,σ2,fμ1logσ1+μ2logσ2μ1logσ1+μ2logσ2ρLlogσ1+logσ2≤μρLloglogMσ1,σ2,fμ1logσ1+μ2logσ2,(75)limsupσ1,σ2→+∞loglogMσ1,σ2,fρ1Llogσ1+ρ2Llogσ2≔α≤μρL≤+∞.
Next, we continue to prove that α=1.
Case 1.
Suppose that α>1; then there exist two constants α1,α2 such that 1<α2<α1<α and (α2-1)ρ1L>ε>0. From (18), there exist two sequences {σ1i}, {σ2j} such that(76)loglogMσ1i,σ2j,fρ1Llogσ1i+ρ2Llogσ2j>α1,i,j=1,2,…that is,(77)Mσ1i,σ2j,f>expexpα1ρ1Llogσ1i+ρ2Llogσ2j,i,j=1,2,…Hence, when j is fixed at j0 and i>i0, we have(78)Mσ1i,σ2j,f>expexpα2ρ1Llogσ1i=expσ1iρ1Lα2,Since (α2-1)ρ1L>ε>0, we can obtain a contradiction with the assumption of Theorem 5 from the above inequality.
Case 2.
Suppose that α<1; then there exist two constants α1,α2 such that α<γ1<γ2<1 and (1-γ2)ρ1L>ε>0. From (75), there exists σ such that for(79)loglogMσ1,σ2,fρ1Llogσ1+ρ2Llogσ2<γ1,for σ1,σ2>σ, that is,(80)Mσ1,σ2,f<expexpγ1ρ1Llogσ1+ρ2Llogσ2,σ1,σ2>σ.Then it follows(81)Mσ1,σ2,f<expexpγ1ρ1Llogσ1+ρ2Llogσ0,σ1≥σ0,0<σ2≤σ0.Thus, we can choose σ∗≥σ0 such that(82)γ1ρ1Llogσ1+ρ2Llogσ2<γ2ρ1Llogσ1,σ1≥σ∗,then for 0<σ2≤σ0, we have(83)Mσ1,σ2,f<expexpγ2ρ1Llogσ1=expσ1ρ1Lγ2.
If σ2>σ0, we can choose σ∗≥σ0 such that σ1≥σ∗ and(84)γ1ρ1Llogσ1+ρ2Llogσ2<γ2ρ1Llogσ1,that is,(85)Mσ1,σ2,f<expexpγ2ρ1Llogσ1=expσ1ρ1Lγ2,as σ1≥σ∗(σ2).
Since ε is arbitrary and (1-γ2)ρ1L>ε>0, from (83) and (85), we can get a contradiction with the assumptions of f(s1,s2) being of finite logarithmic order (ρ1L,ρ2L).
Therefore, this completes the proof of Theroem 5 from Cases 1 and 2.
2.2. The Proof of Theorem 10
The proof of Theorem 10 is similar to the argument as in Theorem 5; in order to facilitate the readers, we still give the proof of Theorem 10 as follows.
From (29), take μ1>τ1L,μ2>τ2L; then(86)limsupσ1,σ2→+∞logMσ1,σ2,fμ1σ1ρ1L+μ2σ2ρ2L≤1.Let μ=maxμ1,μ2,τL=minτ1L,τ2L; then (87)logMσ1,σ2,fτ1Lσ1ρ1L+τ2Lσ2ρ2L≤logMσ1,σ2,fτLσ1ρ1L+σ2ρ2L=logMσ1,σ2,fμ1σ1ρ1L+μ2σ2ρ2Lμ1σ1ρ1L+μ2σ2ρ2LτLσ1ρ1L+σ2ρ2L≤μτLlogMσ1,σ2,fμ1σ1ρ1L+μ2σ2ρ2L.Thus we have(88)limsupσ1,σ2→+∞logMσ1,σ2,fτ1Lσ1ρ1L+τ2Lσ2ρ2L≔A≤μτL≤+∞.Now, we are going to prove that A=1.
Case 1.
Suppose that A>1; then there exist two constants A1,A2 satisfying 1<A2<A1<A and (A2-1)τ1>ε>0. Thus, from (88), there exist two sequences {σ1i},{σ2j} such that(89)logMσ1i,σ2j,fτ1Lσ1iρ1L+τ2Lσ2jρ2L>A1,i,j=1,2,…i.e.,(90)Mσ1i,σ2j,f>expA1τ1Lσ1iρ1L+τ2Lσ2jρ2L,i,j=1,2,…then when j is fixed at j0 and i>i0, it yields(91)Mσ1i,σ2j,f>expA2τ1Lσ1iρ1L=expA2τ1Lσ1iρ1L,Since (A2-1)τ1>ε>0, we can get a contradiction with the assumptions of f(s1,s2) being of finite logarithmic type (τ1L,τ2L).
Case 2.
Suppose that A<1; then there exist two constants A1,A2 satisfying A<A1<A2<1 and (1-A2)τ1>ε>0. From (88), there exists σ such that(92)logMσ1,σ2,fτ1Lσ1ρ1L+τ2Lσ2ρ2L<A1,σ1,σ2>σ,that is,(93)Mσ1,σ2,f<expA1τ1Lσ1ρ1L+τ2Lσ2ρ1L,σ1,σ2>σ.From (93), there exists σ0 such that(94)Mσ1,σ2,f<expA1τ1σ1ρ1L+τ2σ0ρ2L,σ1≥σ0,0<σ2≤σ0.We choose σ∗≥σ0 such that(95)A1τ1Lσ1ρ1L+τ2Lσ2ρ2L<A2τ1Lσ1ρ1L,σ1≥σ∗,Then, for 0<σ2≤σ0, it follows(96)Mσ1,σ2,f<expA2τ1Lσ1ρ1L.
If σ2>σ0, we can choose σ∗≥σ0 such that for σ1≥σ∗(97)A1τ1Lσ1ρ1L+τ2Lσ2ρ2L<A2τ1Lσ1ρ1L,that is,(98)Mσ1,σ2,f<expγ2τ1Lσ1ρ1L,σ1≥σ∗σ2.
Since ε is arbitrary and (1-A2)τ1>ε>0, in view of (96) and (98), it yields a contradiction with the assumption of f(s1,s2) being of finite logarithmic type (τ1L,τ2L).
Therefore, we completes the proof of Theorem 10 from Cases 1 and 2.
2.3. Proofs of Theorems 15 and 22
Similar to the same argument as in the proofs of Theorems 5 and 10, we can complete the proofs of Theorems 15 and 22 easily.
3. Proofs of Theorems 26 and 273.1. The proof of Theorem 26
For a fixed value of σ2>0, let(99)Gσ1,σ2,f=logMσ1,σ2,fσ1,it can easily be shown that G(σ1,σ2) is monotonic increasing for σ1≥σ(1)=σ(1)(f,σ2). Set ζ1 such that Rζ1=σ1 and fζ1,s2=M(σ1,σ2); then we have(100)M1σ1,σ2,f≥fζ1ζ1,s2=limt→0fζ1,s2-fζ1-t,s2t≥limt→0Mσ1,σ2,f-Mσ1-t,σ2,ft=limt→0expσ1Gσ1,σ2,f-expσ1-tGσ1-t,σ2,tt≥limt→0expσ1Gσ1,σ2,f-expσ1-tGσ1,σ2,tt=Gσ1,σ2,fexpσ1Gσ1,σ2,f,it follows(101)M1σ1,σ2,fMσ1,σ2,f≥logMσ1,σ2,fσ1,forσ1≥σ1.Similar, there exists σ(2)>0 such that(102)M2σ1,σ2,fMσ1,σ2,f≥logMσ1,σ2,fσ2,forσ2≥σ2.From (101) and (102), we have(103)logM1σ1,σ2,f-logMσ1,σ2,f≥loglogMσ1,σ2,f-logσ1,forσ1≥σ1.(104)logM2σ1,σ2,f-logMσ1,σ2,f≥loglogMσ1,σ2,f-logσ2,forσ2≥σ2.
Since f(s1,s2) is of finite logarithmic order (ρ1L,ρ2L), 1<ρ1L,ρ2L<+∞, it follows from (103) and (104) that(105)limsupσ2→+∞limsupσ1→+∞logM1σ1,σ2,f-logMσ1,σ2,flogσ1≥ρ1L-1,and(106)limsupσ1→+∞limsupσ2→+∞logM2σ1,σ2,f-logMσ1,σ2,flogσ2≥ρ2L-1.
On the other hand, for any fixed value of σ2>0, logM(σ1,σ2,f) is an increasing convex function of σ1; then(107)logM2σ1,σ2,f=logMσ1,σ2,f+∫σ12σ1∂/∂t1Mt1,σ2,fMt1,σ2,fdt1≥2σ1M1σ1,σ2,fMσ1,σ2,f.Hence, from (107) and the assumptions of Theorem 26, we can easily get(108)limsupσ2→+∞limsupσ1→+∞logM1σ1,σ2,f-logMσ1,σ2,flogσ1≤ρ1L-1,Similarly,(109)limsupσ1→+∞limsupσ2→+∞logM2σ1,σ2,f-logMσ1,σ2,flogσ2≥ρ2L-1.Therefore, the conclusion of Theorem 26 follows from (105), (106), (108), and (109).
Thus, this completes the proof of Theorem 26.
3.2. The Proof of Theorem 27
Since f(s1,s2) is of finite logarithmic order (ρ1L,ρ2L), 1<ρ1L,ρ2L<+∞, from Theorem 26, we have that for any ε>0, and σ2>0, there exists a real number σ(1)≔σ(1)(ε,σ2) such that(110)M1σ1,σ2,fMσ1,σ2,f<expρ1L+εlogσ1,forσ1≥σ1,and for any σ1>0, there exists a real number σ(2)≔σ(2)(ε,σ2) such that(111)M2σ1,σ2,fMσ1,σ2,f<expρ2L+εlogσ2,forσ2≥σ2.Thus, there exists a real number σ=σ(ε)≥maxσ1,σ2 such that(112)M1σ1,σ2,fMσ1,σ2,f+M2σ1,σ2,fMσ1,σ2,f<expρ1L+εlogσ1+ρ2L+εlogσ2,for σ1,σ2>σ. Then it follows(113)limsupσ1,σ2→+∞logM1σ1,σ2,f+M2σ1,σ2,f-logMσ1,σ2,fρ1L-1logσ1+ρ2L-1logσ2≔B≤1.
Here we prove that B=1. If B<1, there exists two constants B<B1<B2<1 such that 0<ε<(ρ1L-1)(1-B2) and(114)M1σ1,σ2,fMσ1,σ2,f+M1σ1,σ2,fMσ1,σ2,f<expB1ρ1L-1logσ1+ρ2L-1logσ2,for σ1,σ2≥σ. Thus, from (114), we obtain that, for any σ2>0, there exists a real number σ(1)≔σ(1)(σ2) such that(115)M1σ1,σ2,fMσ1,σ2,f<expB2ρ1L-1logσ1,forσ1≥σ1.Hence, since 0<ε<(ρ1L-1)(1-B2), from Theorem 26, we can easily obtain a contradiction with the assumption that f(s1,s2) is of finite partial logarithmic order ρ1L with respect to s1.
Thus, B=1; this completes the proof of Theorem 27.
4. Proofs of Theorems 28 and 29
From the definition of h(x), it follows that(116)limx→+∞hcxhx=1,limx→+∞hx+chx=1,for any finite constant c>0. Moreover, we have the following.
Lemma 34 (see [6, 9, 24]).
Let h(x)∈F,p≥2 and φ(x) be the nonconstant function satisfying φ(x)→+∞ as x→+∞ and(117)limsupx→+∞log+φxx=ϱ,0≤ϱ<∞,if M(x) satisfies limsupx→+∞h(logM(x))/x=ν(>0). Then we have(118)limsupx→+∞hφxlogMxx=ν.
Remark 35.
Under the assumptions of Lemma 34, for p=1, we have(119)limsupx→+∞hφxlogMxx=limsupx→+∞hlogMxx=ν,as ϱ=0.
4.1. The Proof of Theorem 28
By using the same argument as in the proof of Theorem 26, and combining with Lemma 34, we can easily prove Theorem 28.
4.2. The Proof of Theorem 29
Since f(s1,s2) is of finite h-order (ρ1h,ρ2h), 0<ρ1h,ρ2h<+∞, from Theorem 28, we have that for any ε>0, and σ2>0, there exists a real number σ(1)≔σ(1)(ε,σ2) such that(120)M1σ1,σ2,fMσ1,σ2,f<γρ1h+εσ1,forσ1≥σ1,and for any σ1>0, there exists a real number σ(2)≔σ(2)(ε,σ2) such that(121)M2σ1,σ2,fMσ1,σ2,f<γρ2h+εσ2,forσ2≥σ2.Since h(x) is strictly increasing and γ(x) is the inverse function of h(x), then γ(x) is also strictly increasing. Thus, there exists a real number σ=σ(ε)≥maxσ1,σ2 such that(122)M1σ1,σ2,fMσ1,σ2,f<γρ1h+εσ1+ρ2h+εσ2,and(123)M2σ1,σ2,fMσ1,σ2,f<γρ1h+εσ1+ρ2h+εσ2,for σ1,σ2>σ. Thus, we have (124)M1σ1,σ2,fMσ1,σ2,f+M2σ1,σ2,fMσ1,σ2,f<2γρ1h+εσ1+ρ2h+εσ2,that is,(125)h12M1σ1,σ2,fMσ1,σ2,f+M2σ1,σ2,fMσ1,σ2,f<ρ1h+εσ1+ρ2h+εσ2for σ1,σ2>σ. By Lemma 34, it follows(126)limsupσ1,σ2→+∞hM1σ1,σ2,f+M2σ1,σ2,f/Mσ1,σ2,fρ1hσ1+ρ2hσ2≔η≤1.
Here is to prove that η=1. If η<1, there exist two constants η<η1<η2<1 such that 0<ε<ρ1h(1-η2) and(127)M1σ1,σ2,fMσ1,σ2,f+M1σ1,σ2,fMσ1,σ2,f<expη1ρ1hσ1+ρ2hσ2,for σ1,σ2≥σ. Thus, from (127), we obtain that, for any σ2>0, there exists a real number σ(1)≔σ(1)(σ2) such that(128)M1σ1,σ2,fMσ1,σ2,f<M1σ1,σ2,fMσ1,σ2,f+M1σ1,σ2,fMσ1,σ2,f<expη2ρ1hσ1,forσ1≥σ1.Hence, since 0<ε<ρ1h(1-η2), from Theorem 28, we can easily obtain a contradiction with the assumption that f(s1,s2) is of finite partial h-order ρ1h with respect to s1.
Thus, η=1; this completes the proof of Theorem 29.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest in the manuscript.
Authors’ Contributions
Hong-Yan Xu contributed to conceptualization; Yong-Qin Cui and Hong-Yan Xu contributed to writing-original draft preparation; Hong-Yan Xu contributed to writing-review and editing; Yong-Qin Cui, Hong-Yan Xu, and Na Li contributed to funding acquisition. All authors read and approved the final manuscript.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant No. 11561033), the Natural Science Foundation of Jiangxi Province in China (20181BAB201001), and the Foundation of Education Department of Jiangxi (GJJ170759, GJJ170788) of China.
YuJ. R.DingX. Q.TianF. J.2004Wuhan, ChinaPress in Wuhan UniversityAkankshaA.SrivastavaG.Spaces of vector-valued Dirichlet series in a half plane2014961239125210.1007/s11464-014-0396-0MR3260996Meshreky-DaoudS.On the class of entire functions defined by Dirichlet series of several complex variables1991162129429910.1016/0022-247X(91)90195-6MR1135279Zbl0760.320052-s2.0-44949274551FilevichP. V.On the Phragmén-Lindelöf indicator for random entire functions2000521016341637GaoZ. S.The growth of entire functions represented by Dirichlet series1999424741748MR1738224Zbl1014.30002HuoY.KongY.On the generalized order of Dirichlet series200535113313910.1016/S0252-9602(14)60146-6MR3283243JinQ. Y.DengG. T.SunD. C.Julia lines of general random Dirichlet series201262491993610.1007/s10587-012-0074-xMR3010248JinQ. Y.KongY. Y.The random Dirichlet series of infinite order dealing with small function201741252255Zbl1389.30005KongY.GanH.On orders and types of Dirichlet series of slow growth2010341111MR2654411LiaoL. W.YangC. C.Some new and old (unsolved) problems and conjectures on factorization theory, dynamics and functional equations of meromorphic functions201741242247Zbl1389.30129RuM.The recent progress in Nevanlinna theory201842111Zbl06961285ShangL.GaoZ.Entire functions defined by Dirichlet series2008339285386210.1016/j.jmaa.2007.06.073MR2375242Zbl1136.300032-s2.0-35348962601SunD. C.GaoZ. S.The growth of the Dirichlet series in the half plane2002224557563MR1942718Zbl1011.30002YuJ. R.Some properties of multiple Taylor series and random Taylor series200626356857610.1016/S0252-9602(06)60082-9MR2244694YuJ.-R.Julia lines of random Dirichlet series2004128534135310.1016/j.bulsci.2004.02.005MR2066344ChangX.LiuS.ZhaoP.LiX.Convergent prediction-correction-based {ADMM} for multi-block separable convex programming201833527028810.1016/j.cam.2017.11.033MR3758646Zbl1397.902992-s2.0-85039805190ChangX.LiuS.ZhaoP.A note on the sufficient initial condition ensuring the convergence of directly extended 3-block {ADMM} for special semidefinite programming201867101729174310.1080/02331934.2018.1490956MR3883000KongY. Y.On some q-orders and q-types of Dirichlet-Hadamard product function200952611651172MR2640624KongY. Y.DengG. T.The Dirichlet-Hadamard product of Dirichlet series2014352145152MR3221961ReddyA. R.Approximation of an entire function1970312813710.1016/0021-9045(70)90020-1MR0259453Zbl0207.073022-s2.0-49849116343ReddyA. R.Best polynomial approximation to certain entire functions1972519711210.1016/0021-9045(74)90048-3MR0404635SheremetaM. N.FedynyakS. I.On the derivative of the Dirichlet series199839118119710.1007/BF02732373MR1623696XuH. Y.YiC. F.An approximation problem for Dirichlet series of finite order in the half plane2010533617624MR2723547XuH. Y.YiC. F.Growth and approximation of Dirichlet series of infinite order20134218188MR3098885Zbl1299.30003YuJ. R.The convergence of two-tuple Dirichet series and two-tuple laplace transform19621116LiuJ.GaoZ. S.Growth of double Dirichlet series2009294958968MR2568918Zbl1212.30006GaoG. N.The growth of double Dirichlet series201124368372LiangM.On order and type of multiple Dirichlet series201535370370810.1016/S0252-9602(15)30015-1MR3334165Zbl1340.32003LiangM.GaoZ.On the convergence and growth of multiple Dirichlet series20108873274010.1134/S000143461011012XMR2868398LiangM.GaoZ.Convergence and growth of multiple Dirichlet series20103051640164810.1016/S0252-9602(10)60157-9MR2778633