# The Growth on the Maximum Modulus of Double Dirichlet Series

Authors: Yong-Qin Cui; Hong-Yan Xu; Na Li
Journal: Journal of Function Spaces
 (2019)

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## Abstract

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.


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## Body

## 1. 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,nexp⁡λms1+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)lim⁡supm+n→+∞⁡log⁡m+nλm+μn=E<+∞,lim⁡supm+n→+∞⁡log⁡am,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.
Supposef(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:(i)
For any fixed value ofσ2>0, there exists σ10=σ10(K1,β1,σ2) such that(9)Mσ1,σ2,f<exp⁡K1σ1β1,forσ1≥σ10,(ii)
For any fixed value ofσ1>0, there exists σ20=σ20(K2,β2,σ1) such that(10)Mσ1,σ2,f<exp⁡K2σ2β2,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<exp⁡K1σ1β1+K2σ2β2,forσ1,σ2≥σ.Definition 2.
Suppose thatf(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<exp⁡σ1ρ1L+ε,forσ1≥σ1,and there exists at least a real value σ20(ε) and sufficiently large number σ1i such that(13)Mσ1i,σ20ε,f>exp⁡σ1iρ1L-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<exp⁡σ2ρ2L+ε,forσ2≥σ2,and there exists at least a real value σ10(ε) and sufficiently large number σ2j such that(15)Mσ2j,σ10ε,f>exp⁡σ2jρ2L-ε,then we say that f(s1,s2) has partial logarithmic order ρ2L with respect to s2.Remark 3.
Iff(s1,s2) has partial logarithmic order ρ1L with respect to s1, that is,(16)lim⁡supσ2→+∞⁡lim⁡supσ1→+∞⁡log⁡log⁡Mσ1,σ2,flog⁡σ1=ρ1L;and if f(s1,s2) has partial logarithmic order ρ2 with respect to s2, that is,(17)lim⁡supσ1→+∞⁡lim⁡supσ2→+∞⁡log⁡log⁡Mσ1,σ2,flog⁡σ2=ρ2L.Definition 4.
Letf(s1,s2)∈D, 0<ρ1L,ρ2L<∞, and satisfy the following conditions:(i)
f(s1,s2)∈L;(ii)
f(s1,s2) has partial logarithmic order ρ1L with respect to s1 and partial logarithmic order ρ2L with respect to s2;(iii)
for any small numberε>0, there exists σ=σ(ε) such that(18)Mσ1,σ2,f<exp⁡σ1ρ1L+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 seriesf(s1,s2), we have the following.Theorem 5.
Letf(s1,s2)∈D be of finite logarithmic order (ρ1L,ρ2L), 1<ρ1L,ρ2L<+∞; then(19)lim⁡supσ1,σ2→+∞⁡log⁡log⁡Mσ1,σ2,fρ1Llog⁡σ1+ρ2Llog⁡σ2=1.To estimate the growth off(s1,s2) more precisely, we will give the logarithmic type of f(s1,s2) as follows.Definition 6.
Supposef(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:(i)
For any fixed value ofσ2>0, there exists σ10=σ10(K1,σ2) such that(20)Mσ1,σ2,f<exp⁡K1ρ1Lσ1,forσ1≥σ10,(ii)
For any fixed value ofσ1>0, there exists σ20=σ20(K2,σ1) such that(21)Mσ1,σ2,f<exp⁡K2ρ2Lσ2,forσ2≥σ20,where K1>0,K2>0, are constants, so there exists σ=σ(K1,K2,) such that(22)Mσ1,σ2,f<exp⁡K1ρ1Lσ1+K2ρ2Lσ2,forσ1,σ2≥σ.Definition 7.
Suppose thatf(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<exp⁡τ1L+ερ1Lσ1,forσ1≥σ1,and there exist at least a real value σ20(ε) and sufficiently large number σ1i such that(24)Mσ1i,σ20ε,f>exp⁡τ1L-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<exp⁡τ2L+ερ2Lσ2,forσ2≥σ2,and there exists at least a real value σ10(ε) and sufficiently large number σ2j such that(26)Mσ2j,σ10ε,f>exp⁡τ2L-ερ2Lσ2j,then we say that f(s1,s2) have partial logarithmic type τ2L with respect to s2.Remark 8.
Iff(s1,s2) is of finite logarithmic order (ρ1L,ρ2L) having partial logarithmic type τ1L with respect to s1, that is,(27)lim⁡supσ2→+∞⁡lim⁡supσ1→+∞⁡log⁡Mσ1,σ2,fρ1Lσ1=τ1L;and if f(s1,s2) has partial logarithmic order ρ2 with respect to s2, that is,(28)lim⁡supσ1→+∞⁡lim⁡supσ2→+∞⁡log⁡Mσ1,σ2,fρ2Lσ2=τ2L.Definition 9.
Letf(s1,s2)∈D be of finite logarithmic order (ρ1L,ρ2L), 0<ρ1L,ρ2L<∞, and satisfy the following conditions:(i)
f(s1,s2)∈LT;(ii)
f(s1,s2) has partial logarithmic type τ1L with respect to s1 and partial logarithmic type τ2L with respect to s2;(iii)
for any small numberε>0, there exists σ=σ(ε) such that(29)Mσ1,σ2,f<exp⁡τ1L+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 seriesf(s1,s2), we have the following.Theorem 10.
Letf(s1,s2)∈D be of finite logarithmic order (ρ1L,ρ2L), 1<ρ1L,ρ2L<+∞, and of finite logarithmic type (τ1L,τ2L); then(30)lim⁡supσ1,σ2→+∞⁡log⁡Mσ1,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 reduceM(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].LetF be the class of all functions h(x) satisfying the following conditions:(i)
h(x) is defined on [a,+∞) and is positive, strictly increasing, and differentiable and tends to +∞ as x→+∞;(ii)
h(x)~log⁡x as x→+∞ for p=1, and limx→+∞⁡d(h(x))/d(logpx)=0, p>1,p∈N+, where log0x=x,log1x=log⁡x and logpx=log⁡(logp-1x).Similar to the above definitions, we give theh-order of double Dirichlet series as follows.Definition 11.
Supposef(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:(i)
For any fixed value ofσ2>0, there exists σ10=σ10(K1,β1,σ2) such that(31)Mσ1,σ2,f<exp⁡γK1σ1β1,forσ1≥σ10,(ii)
For any fixed value ofσ1>0, there exists σ20=σ20(K2,β2,σ1) such that(32)Mσ1,σ2,f<exp⁡γK2σ2β2,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<exp⁡γK1σ1β1+K2σ2β2,forσ1,σ2≥σ,where γ(x) is the inverse function of h(x), especially γ(x)=ex as p=1.Definition 12.
Suppose thatf(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<exp⁡γσ1ρ1h+ε,forσ1≥σ1,and there exist at least a real value σ20(ε) and sufficiently large number σ1i such that(35)Mσ1i,σ20ε,f>exp⁡γσ1iρ1h-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<exp⁡γσ2ρ2h+ε,forσ2≥σ2,and there exist at least a real value σ10(ε) and sufficiently large number σ2j such that(37)Mσ2j,σ10ε,f>exp⁡γσ2jρ2h-ε,then we say that f(s1,s2) has partial h-order ρ2h with respect to s2.Remark 13.
Iff(s1,s2) has partial h-order ρ1h with respect to s1, that is,(38)lim⁡supσ2→+∞⁡lim⁡supσ1→+∞⁡hlog⁡Mσ1,σ2,fσ1=ρ1h;and if f(s1,s2) has partial h-order ρ2h with respect to s2, that is,(39)lim⁡supσ1→+∞⁡lim⁡supσ2→+∞⁡hlog⁡Mσ1,σ2,fσ2=ρ2h.Definition 14.
Letf(s1,s2)∈D, 0<ρ1h,ρ2h<∞, and satisfy the following conditions:(i)
f(s1,s2)∈Lh;(ii)
f(s1,s2) has partial h-order ρ1h with respect to s1 and partial h-order ρ2h with respect to s2;(iii)
for any small numberε>0, there exists σ=σ(ε) such that (40)Mσ1,σ2,f<exp⁡γσ1ρ1h+ε+σ2ρ2h+ε,asσ1,σ2>σ,then we say that f(s1,s2) has finite h-order (ρ1h,ρ2h).Forh-order of double Dirichlet series, we have the following.Theorem 15.
Letf(s1,s2)∈D and be of finite h-order (ρ1h,ρ2h), 0<ρ1h,ρ2h<+∞; then(41)lim⁡supσ1,σ2→+∞⁡hlog⁡Mσ1,σ2,fρ1hσ1+ρ2hσ2=1.In fact, whenh(x)=log⁡(x), then h-order is called finite order. Thus, we obtain the following conclusion.Corollary 16.
Letf(s1,s2)∈D be of finite order (ρ1,ρ2), 0<ρ1,ρ2<∞; then(42)lim⁡supσ1,σ2→+∞⁡log⁡log⁡Mσ1,σ2,fρ1σ1+ρ2σ2=1.And whenh(x)=logpx,p≥2 and p∈N+, then h-order is called the p-order. So, we have the following.Corollary 17.
Letf(s1,s2)∈D be of finite p-order (ρ1p,ρ2), 0<ρ1p,ρ2p<+∞; then(43)lim⁡supσ1,σ2→+∞⁡logp+1⁡Mσ1,σ2,fρ1pσ1+ρ2pσ2=1.Similarly, we give theh-type of double Dirichlet series f(s1,s2) as follows.Definition 18.
Supposef(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:(i)
For any fixed value ofσ2>0, there exists σ10=σ10(K1,β1,σ2) such that(44)Mσ1,σ2,f<γK1exp⁡ρ1hσ1,forσ1≥σ10,(ii)
For any fixed value ofσ1>0, there exists σ20=σ20(K2,β2,σ1) such that(45)Mσ1,σ2,f<γK2exp⁡ρ2hσ2,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<γK1exp⁡ρ1hσ1+K2exp⁡ρ2hσ2,forσ1,σ2≥σ.Definition 19.
Suppose thatf(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+εexp⁡ρ1hσ1,forσ1≥σ1,and there exist at least a real value σ20(ε) and sufficiently large number σ1i such that(48)Mσ1i,σ20ε,f>exp⁡γ(τ1h-εexp⁡ρ1hσ1i,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+εexp⁡ρ2hσ2,forσ2≥σ2,and there exist at least a real value σ10(ε) and sufficiently large number σ2j such that(50)Mσ2j,σ10ε,f>γτ2h-εexp⁡ρ2hσ2i,then we say that f(s1,s2) has partial h-type ρ2h with respect to s2.Remark 20.
Iff(s1,s2) has partial h-type τ1h with respect to s1, that is,(51)lim⁡supσ2→+∞⁡lim⁡supσ1→+∞⁡hMσ1,σ2,fexp⁡ρ1hσ1=τ1h;and if f(s1,s2) has partial h-order ρ2h with respect to s2, that is,(52)lim⁡supσ1→+∞⁡lim⁡supσ2→+∞⁡hMσ1,σ2,fexp⁡ρ2hσ2=τ2h.Definition 21.
Letf(s1,s2)∈D be of finite h-order (ρ1h,ρ2h), 0<ρ1h,ρ2h<+∞, and satisfy the following conditions:(i)
f(s1,s2)∈Lh;(ii)
f(s1,s2) has partial h-type τ1h with respect to s1 and partial h-type τ2h with respect to s2;(iii)
for any small numberε>0, there exists σ=σ(ε) such that (53)Mσ1,σ2,f<γτ1h+εexp⁡ρ1hσ1+τ2h+εexp⁡ρ2hσ2,asσ1,σ2>σ,then we say that f(s1,s2) has finite h-type (τ1h,τ2h).Forh-type of double Dirichlet series f(s1,s2), we have the following.Theorem 22.
Letf(s1,s2)∈D and be of finite h-order (ρ1h,ρ2h), 0<ρ1h,ρ2h<+∞, and of finite h-type (τ1h,τ2h); then(54)lim⁡supσ1,σ2→+∞⁡hMσ1,σ2,fτ1hexp⁡ρ1hσ1+τ2hexp⁡ρ2hσ2=1.Particularly, we can get the following corollaries.Corollary 23.
Letf(s1,s2)∈D and be of finite order (ρ1,ρ2) and of finite type (τ1,τ2); then(55)lim⁡supσ1,σ2→+∞⁡log⁡Mσ1,σ2,fτ1exp⁡ρ1σ1+τ2exp⁡ρ2σ2=1.Corollary 24.
Letf(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)lim⁡supσ1,σ2→+∞⁡logp⁡Mσ1,σ2,fτ1pexp⁡ρ1pσ1+τ2pexp⁡ρ2pσ2=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=lim⁡supσ1,σ2→+∞⁡log⁡log⁡Mσ1,σ2,flog⁡eσ1+eσ2,and if ρ∈(0,+∞), then the type of multiple Dirichlet series f(s1,s2)(58)T=lim⁡supσ1,σ2→+∞⁡log⁡Mσ1,σ2,flog⁡eρσ1+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 derivativesfsj(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.
Letf(s1,s2)∈D be of finite logarithmic order (ρ1L,ρ2L), 1<ρ1L,ρ2L<+∞; then(61)lim⁡supσ2→∞⁡lim⁡supσ1→+∞⁡log⁡M1σ1,σ2,f-log⁡Mσ1,σ2,flog⁡σ1=ρ1L-1,and(62)lim⁡supσ1→+∞⁡lim⁡supσ2→+∞⁡log⁡M2σ1,σ2,f-log⁡Mσ1,σ2,flog⁡σ2=ρ2L-1.Theorem 27.
Letf(s1,s2)∈D be of finite logarithmic order (ρ1L,ρ2L), 1<ρ1L,ρ2L<+∞; then(63)lim⁡supσ1,σ2→+∞⁡log⁡M1σ1,σ2,f+M2σ1,σ2,f-log⁡Mσ1,σ2,fρ1L-1log⁡σ1+ρ2L-1log⁡σ2=1.Theorem 28.
Letf(s1,s2)∈D be of finite h-order (ρ1h,ρ2h), 0<ρ1h,ρ2h<+∞; then(64)lim⁡supσ2→+∞⁡lim⁡supσ1→+∞⁡hM1σ1,σ2,f/Mσ1,σ2,fσ1=ρ1h,and(65)lim⁡supσ1→+∞⁡lim⁡supσ2→+∞⁡hM2σ1,σ2,f/Mσ1,σ2,fσ2=ρ2h.Theorem 29.
Letf(s1,s2)∈D be of finite h-order (ρ1h,ρ2h), 0<ρ1h,ρ2h<+∞; then(66)lim⁡supσ1,σ2→+∞⁡hM1σ1,σ2,f+M2σ1,σ2,f/Mσ1,σ2,fρ1hσ1+ρ2hσ2=1.For finite order and finitep-order, some corollaries are obtained below.Corollary 30.
Letf(s1,s2)∈D be of finite order (ρ1,ρ2), 0<ρ1,ρ2<+∞; then(67)lim⁡supσ2→+∞⁡lim⁡supσ1→+∞⁡log⁡M1σ1,σ2,f-log⁡Mσ1,σ2,fσ1=ρ1,and(68)lim⁡supσ2→+∞⁡lim⁡supσ1→+∞⁡log⁡M2σ1,σ2,f-logMσ1,σ2,fσ2=ρ2.Corollary 31.
Letf(s1,s2)∈D be of finite order (ρ1,ρ2), 0<ρ1,ρ2<+∞; then(69)lim⁡supσ1,σ2→+∞⁡log⁡M1σ1,σ2,f+M2σ1,σ2,f-log⁡Mσ1,σ2,fρ1σ1+ρ2σ2=1.Corollary 32.
Letf(s1,s2)∈D be of finite p-order (ρ1p,ρ2p), 0<ρ1p,ρ2p<+∞, p≥2; then(70)lim⁡supσ2→+∞⁡lim⁡supσ1→+∞⁡logp⁡M1σ1,σ2,f/Mσ1,σ2,fσ1=ρ1p,and(71)lim⁡supσ2→+∞⁡lim⁡supσ1→+∞⁡logp⁡M2σ1,σ2,f/Mσ1,σ2,fσ2=ρ2p.Corollary 33.
Letf(s1,s2)∈D be of finite p-order (ρ1p,ρ2p), 0<ρ1p,ρ2p<+∞,p≥2; then(72)lim⁡supσ1,σ2→+∞⁡logp⁡M1σ1,σ2,f+M2σ1,σ2,f/Mσ1,σ2,fρ1pσ1+ρ2pσ2=1.


## 2. Proofs of Theorems5–22

### 2.1. The Proof of Theorem5

From (18), let μ1>ρ1L,μ2>ρ2L; it follows(73)lim⁡supσ1,σ2→+∞⁡log⁡log⁡Mσ1,σ2μ1log⁡σ1+μ2log⁡σ2≤1.When σ1>0,σ2>0, let μ=max⁡μ1,μ2,ρL=min⁡ρ1L,ρ2L; then (74)log⁡log⁡Mσ1,σ2,fρ1Llog⁡σ1+ρ2Llog⁡σ2≤log⁡log⁡Mσ1,σ2,fρLlog⁡σ1+log⁡σ2=log⁡log⁡Mσ1,σ2,fμ1log⁡σ1+μ2log⁡σ2μ1log⁡σ1+μ2log⁡σ2ρLlog⁡σ1+log⁡σ2≤μρLlog⁡log⁡Mσ1,σ2,fμ1log⁡σ1+μ2log⁡σ2,(75)lim⁡supσ1,σ2→+∞⁡log⁡log⁡Mσ1,σ2,fρ1Llog⁡σ1+ρ2Llog⁡σ2≔α≤μρL≤+∞.Next, we continue to prove thatα=1.Case 1.
Suppose thatα>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)log⁡log⁡Mσ1i,σ2j,fρ1Llog⁡σ1i+ρ2Llog⁡σ2j>α1,i,j=1,2,…that is,(77)Mσ1i,σ2j,f>exp⁡exp⁡α1ρ1Llog⁡σ1i+ρ2Llog⁡σ2j,i,j=1,2,…Hence, when j is fixed at j0 and i>i0, we have(78)Mσ1i,σ2j,f>exp⁡exp⁡α2ρ1Llog⁡σ1i=exp⁡σ1iρ1Lα2,Since (α2-1)ρ1L>ε>0, we can obtain a contradiction with the assumption of Theorem 5 from the above inequality.Case 2.
Suppose thatα<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)log⁡log⁡Mσ1,σ2,fρ1Llog⁡σ1+ρ2Llog⁡σ2<γ1,for σ1,σ2>σ, that is,(80)Mσ1,σ2,f<exp⁡exp⁡γ1ρ1Llog⁡σ1+ρ2Llog⁡σ2,σ1,σ2>σ.Then it follows(81)Mσ1,σ2,f<exp⁡exp⁡γ1ρ1Llog⁡σ1+ρ2Llog⁡σ0,σ1≥σ0,0<σ2≤σ0.Thus, we can choose σ∗≥σ0 such that(82)γ1ρ1Llog⁡σ1+ρ2Llog⁡σ2<γ2ρ1Llog⁡σ1,σ1≥σ∗,then for 0<σ2≤σ0, we have(83)Mσ1,σ2,f<exp⁡exp⁡γ2ρ1Llog⁡σ1=exp⁡σ1ρ1Lγ2.
Ifσ2>σ0, we can choose σ∗≥σ0 such that σ1≥σ∗ and(84)γ1ρ1Llog⁡σ1+ρ2Llog⁡σ2<γ2ρ1Llog⁡σ1,that is,(85)Mσ1,σ2,f<exp⁡exp⁡γ2ρ1Llog⁡σ1=exp⁡σ1ρ1Lγ2,as σ1≥σ∗(σ2).
Sinceε 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 Theroem5 from Cases 1 and 2.


### 2.2. The Proof of Theorem10

The proof of Theorem10 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)lim⁡supσ1,σ2→+∞⁡log⁡Mσ1,σ2,fμ1σ1ρ1L+μ2σ2ρ2L≤1.Let μ=max⁡μ1,μ2,τL=min⁡τ1L,τ2L; then (87)log⁡Mσ1,σ2,fτ1Lσ1ρ1L+τ2Lσ2ρ2L≤log⁡Mσ1,σ2,fτLσ1ρ1L+σ2ρ2L=log⁡Mσ1,σ2,fμ1σ1ρ1L+μ2σ2ρ2Lμ1σ1ρ1L+μ2σ2ρ2LτLσ1ρ1L+σ2ρ2L≤μτLlog⁡Mσ1,σ2,fμ1σ1ρ1L+μ2σ2ρ2L.Thus we have(88)lim⁡supσ1,σ2→+∞⁡log⁡Mσ1,σ2,fτ1Lσ1ρ1L+τ2Lσ2ρ2L≔A≤μτL≤+∞.Now, we are going to prove that A=1.Case 1.
Suppose thatA>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)log⁡Mσ1i,σ2j,fτ1Lσ1iρ1L+τ2Lσ2jρ2L>A1,i,j=1,2,…i.e.,(90)Mσ1i,σ2j,f>exp⁡A1τ1Lσ1iρ1L+τ2Lσ2jρ2L,i,j=1,2,…then when j is fixed at j0 and i>i0, it yields(91)Mσ1i,σ2j,f>exp⁡A2τ1Lσ1iρ1L=exp⁡A2τ1Lσ1iρ1L,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 thatA<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)log⁡Mσ1,σ2,fτ1Lσ1ρ1L+τ2Lσ2ρ2L<A1,σ1,σ2>σ,that is,(93)Mσ1,σ2,f<exp⁡A1τ1Lσ1ρ1L+τ2Lσ2ρ1L,σ1,σ2>σ.From (93), there exists σ0 such that(94)Mσ1,σ2,f<exp⁡A1τ1σ1ρ1L+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<exp⁡A2τ1Lσ1ρ1L.
Ifσ2>σ0, we can choose σ∗≥σ0 such that for σ1≥σ∗(97)A1τ1Lσ1ρ1L+τ2Lσ2ρ2L<A2τ1Lσ1ρ1L,that is,(98)Mσ1,σ2,f<exp⁡γ2τ1Lσ1ρ1L,σ1≥σ∗σ2.
Sinceε 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 Theorem10 from Cases 1 and 2.


### 2.3. Proofs of Theorems15 and 22

Similar to the same argument as in the proofs of Theorems5 and 10, we can complete the proofs of Theorems 15 and 22 easily.


## 2.1. The Proof of Theorem5

From (18), let μ1>ρ1L,μ2>ρ2L; it follows(73)lim⁡supσ1,σ2→+∞⁡log⁡log⁡Mσ1,σ2μ1log⁡σ1+μ2log⁡σ2≤1.When σ1>0,σ2>0, let μ=max⁡μ1,μ2,ρL=min⁡ρ1L,ρ2L; then (74)log⁡log⁡Mσ1,σ2,fρ1Llog⁡σ1+ρ2Llog⁡σ2≤log⁡log⁡Mσ1,σ2,fρLlog⁡σ1+log⁡σ2=log⁡log⁡Mσ1,σ2,fμ1log⁡σ1+μ2log⁡σ2μ1log⁡σ1+μ2log⁡σ2ρLlog⁡σ1+log⁡σ2≤μρLlog⁡log⁡Mσ1,σ2,fμ1log⁡σ1+μ2log⁡σ2,(75)lim⁡supσ1,σ2→+∞⁡log⁡log⁡Mσ1,σ2,fρ1Llog⁡σ1+ρ2Llog⁡σ2≔α≤μρL≤+∞.Next, we continue to prove thatα=1.Case 1.
Suppose thatα>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)log⁡log⁡Mσ1i,σ2j,fρ1Llog⁡σ1i+ρ2Llog⁡σ2j>α1,i,j=1,2,…that is,(77)Mσ1i,σ2j,f>exp⁡exp⁡α1ρ1Llog⁡σ1i+ρ2Llog⁡σ2j,i,j=1,2,…Hence, when j is fixed at j0 and i>i0, we have(78)Mσ1i,σ2j,f>exp⁡exp⁡α2ρ1Llog⁡σ1i=exp⁡σ1iρ1Lα2,Since (α2-1)ρ1L>ε>0, we can obtain a contradiction with the assumption of Theorem 5 from the above inequality.Case 2.
Suppose thatα<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)log⁡log⁡Mσ1,σ2,fρ1Llog⁡σ1+ρ2Llog⁡σ2<γ1,for σ1,σ2>σ, that is,(80)Mσ1,σ2,f<exp⁡exp⁡γ1ρ1Llog⁡σ1+ρ2Llog⁡σ2,σ1,σ2>σ.Then it follows(81)Mσ1,σ2,f<exp⁡exp⁡γ1ρ1Llog⁡σ1+ρ2Llog⁡σ0,σ1≥σ0,0<σ2≤σ0.Thus, we can choose σ∗≥σ0 such that(82)γ1ρ1Llog⁡σ1+ρ2Llog⁡σ2<γ2ρ1Llog⁡σ1,σ1≥σ∗,then for 0<σ2≤σ0, we have(83)Mσ1,σ2,f<exp⁡exp⁡γ2ρ1Llog⁡σ1=exp⁡σ1ρ1Lγ2.
Ifσ2>σ0, we can choose σ∗≥σ0 such that σ1≥σ∗ and(84)γ1ρ1Llog⁡σ1+ρ2Llog⁡σ2<γ2ρ1Llog⁡σ1,that is,(85)Mσ1,σ2,f<exp⁡exp⁡γ2ρ1Llog⁡σ1=exp⁡σ1ρ1Lγ2,as σ1≥σ∗(σ2).
Sinceε 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 Theroem5 from Cases 1 and 2.


## 2.2. The Proof of Theorem10

The proof of Theorem10 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)lim⁡supσ1,σ2→+∞⁡log⁡Mσ1,σ2,fμ1σ1ρ1L+μ2σ2ρ2L≤1.Let μ=max⁡μ1,μ2,τL=min⁡τ1L,τ2L; then (87)log⁡Mσ1,σ2,fτ1Lσ1ρ1L+τ2Lσ2ρ2L≤log⁡Mσ1,σ2,fτLσ1ρ1L+σ2ρ2L=log⁡Mσ1,σ2,fμ1σ1ρ1L+μ2σ2ρ2Lμ1σ1ρ1L+μ2σ2ρ2LτLσ1ρ1L+σ2ρ2L≤μτLlog⁡Mσ1,σ2,fμ1σ1ρ1L+μ2σ2ρ2L.Thus we have(88)lim⁡supσ1,σ2→+∞⁡log⁡Mσ1,σ2,fτ1Lσ1ρ1L+τ2Lσ2ρ2L≔A≤μτL≤+∞.Now, we are going to prove that A=1.Case 1.
Suppose thatA>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)log⁡Mσ1i,σ2j,fτ1Lσ1iρ1L+τ2Lσ2jρ2L>A1,i,j=1,2,…i.e.,(90)Mσ1i,σ2j,f>exp⁡A1τ1Lσ1iρ1L+τ2Lσ2jρ2L,i,j=1,2,…then when j is fixed at j0 and i>i0, it yields(91)Mσ1i,σ2j,f>exp⁡A2τ1Lσ1iρ1L=exp⁡A2τ1Lσ1iρ1L,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 thatA<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)log⁡Mσ1,σ2,fτ1Lσ1ρ1L+τ2Lσ2ρ2L<A1,σ1,σ2>σ,that is,(93)Mσ1,σ2,f<exp⁡A1τ1Lσ1ρ1L+τ2Lσ2ρ1L,σ1,σ2>σ.From (93), there exists σ0 such that(94)Mσ1,σ2,f<exp⁡A1τ1σ1ρ1L+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<exp⁡A2τ1Lσ1ρ1L.
Ifσ2>σ0, we can choose σ∗≥σ0 such that for σ1≥σ∗(97)A1τ1Lσ1ρ1L+τ2Lσ2ρ2L<A2τ1Lσ1ρ1L,that is,(98)Mσ1,σ2,f<exp⁡γ2τ1Lσ1ρ1L,σ1≥σ∗σ2.
Sinceε 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 Theorem10 from Cases 1 and 2.


## 2.3. Proofs of Theorems15 and 22

Similar to the same argument as in the proofs of Theorems5 and 10, we can complete the proofs of Theorems 15 and 22 easily.


## 3. Proofs of Theorems26 and 27

### 3.1. The proof of Theorem26

For a fixed value ofσ2>0, let(99)Gσ1,σ2,f=log⁡Mσ1,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→0⁡fζ1,s2-fζ1-t,s2t≥limt→0⁡Mσ1,σ2,f-Mσ1-t,σ2,ft=limt→0⁡exp⁡σ1Gσ1,σ2,f-exp⁡σ1-tGσ1-t,σ2,tt≥limt→0⁡exp⁡σ1Gσ1,σ2,f-exp⁡σ1-tGσ1,σ2,tt=Gσ1,σ2,fexp⁡σ1Gσ1,σ2,f,it follows(101)M1σ1,σ2,fMσ1,σ2,f≥log⁡Mσ1,σ2,fσ1,forσ1≥σ1.Similar, there exists σ(2)>0 such that(102)M2σ1,σ2,fMσ1,σ2,f≥log⁡Mσ1,σ2,fσ2,forσ2≥σ2.From (101) and (102), we have(103)log⁡M1σ1,σ2,f-log⁡Mσ1,σ2,f≥log⁡log⁡Mσ1,σ2,f-log⁡σ1,forσ1≥σ1.(104)log⁡M2σ1,σ2,f-log⁡Mσ1,σ2,f≥log⁡log⁡Mσ1,σ2,f-log⁡σ2,forσ2≥σ2.Sincef(s1,s2) is of finite logarithmic order (ρ1L,ρ2L), 1<ρ1L,ρ2L<+∞, it follows from (103) and (104) that(105)lim⁡supσ2→+∞⁡lim⁡supσ1→+∞⁡log⁡M1σ1,σ2,f-log⁡Mσ1,σ2,flog⁡σ1≥ρ1L-1,and(106)lim⁡supσ1→+∞⁡lim⁡supσ2→+∞⁡log⁡M2σ1,σ2,f-log⁡Mσ1,σ2,flog⁡σ2≥ρ2L-1.On the other hand, for any fixed value ofσ2>0, log⁡M(σ1,σ2,f) is an increasing convex function of σ1; then(107)log⁡M2σ1,σ2,f=log⁡Mσ1,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)lim⁡supσ2→+∞⁡lim⁡supσ1→+∞⁡log⁡M1σ1,σ2,f-log⁡Mσ1,σ2,flog⁡σ1≤ρ1L-1,Similarly,(109)lim⁡supσ1→+∞⁡lim⁡supσ2→+∞⁡log⁡M2σ1,σ2,f-log⁡Mσ1,σ2,flog⁡σ2≥ρ2L-1.Therefore, the conclusion of Theorem 26 follows from (105), (106), (108), and (109).Thus, this completes the proof of Theorem26.


### 3.2. The Proof of Theorem27

Sincef(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<exp⁡ρ1L+εlog⁡σ1,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<exp⁡ρ2L+εlog⁡σ2,forσ2≥σ2.Thus, there exists a real number σ=σ(ε)≥max⁡σ1,σ2 such that(112)M1σ1,σ2,fMσ1,σ2,f+M2σ1,σ2,fMσ1,σ2,f<exp⁡ρ1L+εlog⁡σ1+ρ2L+εlog⁡σ2,for σ1,σ2>σ. Then it follows(113)lim⁡supσ1,σ2→+∞⁡log⁡M1σ1,σ2,f+M2σ1,σ2,f-log⁡Mσ1,σ2,fρ1L-1log⁡σ1+ρ2L-1log⁡σ2≔B≤1.Here we prove thatB=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<exp⁡B1ρ1L-1log⁡σ1+ρ2L-1log⁡σ2,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<exp⁡B2ρ1L-1log⁡σ1,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.


## 3.1. The proof of Theorem26

For a fixed value ofσ2>0, let(99)Gσ1,σ2,f=log⁡Mσ1,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→0⁡fζ1,s2-fζ1-t,s2t≥limt→0⁡Mσ1,σ2,f-Mσ1-t,σ2,ft=limt→0⁡exp⁡σ1Gσ1,σ2,f-exp⁡σ1-tGσ1-t,σ2,tt≥limt→0⁡exp⁡σ1Gσ1,σ2,f-exp⁡σ1-tGσ1,σ2,tt=Gσ1,σ2,fexp⁡σ1Gσ1,σ2,f,it follows(101)M1σ1,σ2,fMσ1,σ2,f≥log⁡Mσ1,σ2,fσ1,forσ1≥σ1.Similar, there exists σ(2)>0 such that(102)M2σ1,σ2,fMσ1,σ2,f≥log⁡Mσ1,σ2,fσ2,forσ2≥σ2.From (101) and (102), we have(103)log⁡M1σ1,σ2,f-log⁡Mσ1,σ2,f≥log⁡log⁡Mσ1,σ2,f-log⁡σ1,forσ1≥σ1.(104)log⁡M2σ1,σ2,f-log⁡Mσ1,σ2,f≥log⁡log⁡Mσ1,σ2,f-log⁡σ2,forσ2≥σ2.Sincef(s1,s2) is of finite logarithmic order (ρ1L,ρ2L), 1<ρ1L,ρ2L<+∞, it follows from (103) and (104) that(105)lim⁡supσ2→+∞⁡lim⁡supσ1→+∞⁡log⁡M1σ1,σ2,f-log⁡Mσ1,σ2,flog⁡σ1≥ρ1L-1,and(106)lim⁡supσ1→+∞⁡lim⁡supσ2→+∞⁡log⁡M2σ1,σ2,f-log⁡Mσ1,σ2,flog⁡σ2≥ρ2L-1.On the other hand, for any fixed value ofσ2>0, log⁡M(σ1,σ2,f) is an increasing convex function of σ1; then(107)log⁡M2σ1,σ2,f=log⁡Mσ1,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)lim⁡supσ2→+∞⁡lim⁡supσ1→+∞⁡log⁡M1σ1,σ2,f-log⁡Mσ1,σ2,flog⁡σ1≤ρ1L-1,Similarly,(109)lim⁡supσ1→+∞⁡lim⁡supσ2→+∞⁡log⁡M2σ1,σ2,f-log⁡Mσ1,σ2,flog⁡σ2≥ρ2L-1.Therefore, the conclusion of Theorem 26 follows from (105), (106), (108), and (109).Thus, this completes the proof of Theorem26.


## 3.2. The Proof of Theorem27

Sincef(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<exp⁡ρ1L+εlog⁡σ1,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<exp⁡ρ2L+εlog⁡σ2,forσ2≥σ2.Thus, there exists a real number σ=σ(ε)≥max⁡σ1,σ2 such that(112)M1σ1,σ2,fMσ1,σ2,f+M2σ1,σ2,fMσ1,σ2,f<exp⁡ρ1L+εlog⁡σ1+ρ2L+εlog⁡σ2,for σ1,σ2>σ. Then it follows(113)lim⁡supσ1,σ2→+∞⁡log⁡M1σ1,σ2,f+M2σ1,σ2,f-log⁡Mσ1,σ2,fρ1L-1log⁡σ1+ρ2L-1log⁡σ2≔B≤1.Here we prove thatB=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<exp⁡B1ρ1L-1log⁡σ1+ρ2L-1log⁡σ2,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<exp⁡B2ρ1L-1log⁡σ1,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 Theorems28 and 29

From the definition ofh(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]).
Leth(x)∈F,p≥2 and φ(x) be the nonconstant function satisfying φ(x)→+∞ as x→+∞ and(117)lim⁡supx→+∞⁡log+⁡φxx=ϱ,0≤ϱ<∞,if M(x) satisfies lim⁡supx→+∞⁡h(log⁡M(x))/x=ν(>0). Then we have(118)lim⁡supx→+∞⁡hφxlog⁡Mxx=ν.Remark 35.
Under the assumptions of Lemma34, for p=1, we have(119)lim⁡supx→+∞⁡hφxlog⁡Mxx=lim⁡supx→+∞⁡hlog⁡Mxx=ν,as ϱ=0.


### 4.1. The Proof of Theorem28

By using the same argument as in the proof of Theorem26, and combining with Lemma 34, we can easily prove Theorem 28.


### 4.2. The Proof of Theorem29

Sincef(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)≥max⁡σ1,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)lim⁡supσ1,σ2→+∞⁡hM1σ1,σ2,f+M2σ1,σ2,f/Mσ1,σ2,fρ1hσ1+ρ2hσ2≔η≤1.Here is to prove thatη=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<exp⁡η1ρ1hσ1+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<exp⁡η2ρ1hσ1,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.


## 4.1. The Proof of Theorem28

By using the same argument as in the proof of Theorem26, and combining with Lemma 34, we can easily prove Theorem 28.


## 4.2. The Proof of Theorem29

Sincef(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)≥max⁡σ1,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)lim⁡supσ1,σ2→+∞⁡hM1σ1,σ2,f+M2σ1,σ2,f/Mσ1,σ2,fρ1hσ1+ρ2hσ2≔η≤1.Here is to prove thatη=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<exp⁡η1ρ1hσ1+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<exp⁡η2ρ1hσ1,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.



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*Source: 9191346-2019-01-10.xml*