On the limits of real-valued functions in sets involving Ψ-density, and applications

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Rinnakkaistallenteet Luonnontieteiden ja metsätieteiden tiedekunta

2021

On the limits of real-valued functions in

þÿsets involving ¨-density, and applications

Heittokangas, Janne

Elsevier BV

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CC BY http://creativecommons.org/licenses/by/4.0/

http://dx.doi.org/10.1016/j.jmaa.2021.125787

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Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

On the limits of real-valued functions in sets involving ψ-density, and applications

Janne Heittokangas^a, Zinelaabidine Latreuch^b,Jun Wang^c, Amine Zemirni^a,∗

a UniversityofEasternFinland,DepartmentofPhysicsandMathematics,P.O.Box111,80101Joensuu, Finland

bUniversityofMostaganem,DepartmentofMathematics,LaboratoryofPureandAppliedMathematics, B.P.227,Mostaganem,Algeria

cFudanUniversity,SchoolofMathematicalSciences,Shanghai200433,PRChina

a r t i cl e i n f o a b s t r a c t

Articlehistory:

Received30November2020 Availableonline25October2021 Submittedby V.Andriyevskyy

Keywords:

Exceptionalsets Limitoffunctions Meromorphicfunctions Orderofgrowth ψ-Density

Newresultsonupperandlowerlimitsofreal-valuedfunctionsareprovedbymeans of ψ-densitiesintroduced byP. D. Barry in1962.This leadstoimprovements of severalexistingresultsonthegrowthofnon-decreasingandunboundedreal-valued functionsinsetsofpositive density.Theψ-densitiesallowusto introduce anew conceptofalimitforreal-valuedfunctions,whichisusedtorevealfurtherproperties onthebehaviorofintegrablefunctionsatinfinity.

©2021TheAuthor(s).PublishedbyElsevierInc.Thisisanopenaccessarticle undertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Forareal-valued function,thepassage from thelower/upperlimitto theusual limitis validthrougha sequence.Inmanycases,theset,wheretheaforementionedpassage isvalid,turns tobe largerthanjusta sequence.Infact,theproblemoffindingthesizeofthissetisveryinterestinginthetheoryofmeromorphic functions,where the order andlower order of growth areinvolved. Recallthatorder and lower order ofa non-decreasingandunboundedfunctionT,aregiven, respectively,by

ρ(T) = lim sup

r→∞

logT(r)

logr and ρ(T) = lim inf

r→∞

logT(r) logr .

Clearlyρ(T)≤ρ(T) holdsingeneral.Ifsopreferred,T(r) canbereplaced,forexample,withtheNevanlinna characteristicT(r,f) ofameromorphicfunctionf.Theorderρ(f) andthelowerorderρ(f) off aredefined

* Correspondingauthor.

E-mailaddresses:janne.heittokangas@uef.fi(J. Heittokangas),z.latreuch@gmail.com(Z. Latreuch),majwang@fudan.edu.cn (J. Wang),amine.zemirni@uef.fi(A. Zemirni).

https://doi.org/10.1016/j.jmaa.2021.125787

0022-247X/©2021TheAuthor(s). PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

byρ(f)=ρ(T(r,f)) andρ(f)=ρ(T(r,f)).Forentirefunctions,T(r,f) canbereplacedwiththelogarithmic maximum moduluslogM(r,f).

It isknownthat,forany fixedandLsatisfying0≤≤L≤ ∞thereexists ameromorphicfunctionf of order ρ(f)=L andoflower order ρ(f)= [2, p. 238].If< L,then solvingtheaboveproblemallows us toknowthesize ofsetsofr-values,whereT(r,f) isnearmaximal or nearminimal.Relatedtothisend, werecallTheoremAfrom[9,Corollary 3.7],wherethesizeofsuchsetsD⊂[1,∞) ismeasuredintermsof upperand lower(linear)densitiesgivenby

dens(D) = lim sup

r→∞

1 r

D∩[1,r]

dt and dens(D) = lim inf

r→∞

1 r

D∩[1,r]

dt.

TheoremA.Letf beameromorphicfunctionsuchthat 0≤ρ(f)< ρ(f)≤ ∞,andletρ(f)< a≤b< ρ(f).

Then thesets

E={r≥1 :T(r, f)≤r^a} and F ={r≥1 :T(r, f)> r^b} are of upperdensityone andoflowerdensity zero.

Due to the logarithms appearing in the definitions of orders, it seems natural to study these growth questions intermsof logarithmic densities. Recallthat theupper andlower logarithmic densities ofa set D⊂[1,∞) aregivenby

logdens(D) = lim sup

r→∞

1 logr

D∩[1,r]

dt

t and logdens(D) = lim sup

r→∞

1 logr

D∩[1,r]

dt t .

Theconnectionbetweenlinearandlogarithmicdensitiesisapparent fromtheinequalities

0≤dens(D)≤log dens(D)≤log dens(D)≤dens(D)≤1, (1.1) whichcanbefoundin[13,p. 121].

The growth questions above are the motivation to study the limit questions for real-valued functions of arbitrary form.For example,the order and lower order off arethe upper limitand thelower limit of a function ϕ(r) of the particular form logT(r,f)/logr. This calls for afurtherextension forthe concept of density.

We willmake useof generalψ-densitiesintroduced byP. D. Barry [1] in 1962. Thedefinitionsofthese densitiesandsomeoftheirnewconsequencesarediscussedinSection2.Thisallowsustodiscusstheupper andlowerlimitsofarbitraryfunctionsϕ(r) intermsofψ-densitiesinSection3.Consequently,resultsonthe growth of unboundedfunctionsT improvingTheorem A aswell as manyother resultsfrom theliterature are then obtainedas corollariesinSection4. Theψ-densities arenotrestricted to growth questionsalone but they allow us to introduce a new concept of a limit inψ-density for real-valued functionsalso. This extends the concept of a limit in density (statistical convergence), and is used to study the behavior of integrablefunctionsatinfinity.Section5isdevotedtopresentingnewresultsinthisdirection.

2. Barry’sψ-densities

Let0< r₀< R≤ ∞,andletD(r₀,R) denotetheclassofpositive,unbounded,differentiableandstrictly increasing functionson(r₀,R).Letψ∈ D(r₀,R). Thentheψ-measure ofaset E⊂(r₀,R) is thevalueof theintegral

E

dψ(t) =

E

ψ(t)dt.

FollowingBarry[1],theupperandlowerψ-densitiesofE⊂(r0,R) aredefined,respectively, by ψ-dens(E) := lim sup

r→R⁻

1 ψ(r)

E∩[r0,r)

dψ(t) and ψ-dens(E) := lim inf

r→R⁻

1 ψ(r)

E∩[r0,r)

dψ(t).

Itisclearthat0≤ψ-dens(E)≤ψ-dens(E)≤1 and

ψ-dens(E^c) +ψ-dens(E) = 1, (2.1)

whereE^cdenotesthecomplementofEin(r0,R).Generalizingtheinequalitiesin(1.1),theψ-densitiesand thee^ψ-densitiesobeytheinequalities

0≤e^ψ-dens(E)≤ψ-dens(E)≤ψ-dens(E)≤e^ψ-dens(E)≤1 (2.2) foranyset E⊂(r0,R) andforany ψ∈ D(r0,R) by[1,Lemma 1].

Thefollowing special casesare certainly ofinterest. If R= +∞and ψ(r)= logr, then theψ-densities reduceto thelogarithmic densities, whilethe e^ψ-densities coincidewith thelineardensities. IfR = 1 and ψ(r)=−log(1−r),thentheψ-densitiesreduceto thelogarithmicdensities.

The following lemma offers a relation between the ψ-measure and the e^ψ-density, and complements a knownresultforthelogarithmicmeasure [16,p. 9].

Lemma2.1.IfasetE ⊂(r0,R)satisfies

Edψ(t)<∞forψ∈ D(r0,R),thene^ψ-dens(E)= 0.Inparticular, asetE ⊂(r₀,R)of finitelogarithmicmeasure isofzero upperlinear density.

Proof. LetχE(t) bethecharacteristicfunctionoftheset E,andletv(r)=ψ⁻¹(¹₂ψ(r)).Sinceψisstrictly increasing,v(r) iswell-definedandsatisfies v(r)< rforallr > r₀.Then

r r0

χE(t)de^ψ(t)=

v(r)

r0

χE(t)de^ψ(t)+ r v(r)

χE(t)de^ψ(t)

≤

v(r)

r0

de^ψ(t)+e^ψ(r) r v(r)

χ_E(t)dψ(t)

≤e^12ψ(r)+e^ψ(r) r v(r)

χE(t)dψ(t).

Asr→R⁻,we havev(r)→R⁻ andhencer

v(r)χ_E(t)dψ(t)→0.Thus,

e^ψ-dens(E) = lim sup

r→R⁻ e^−ψ(r) r r0

χ_E(t)de^ψ(t)= 0.

Thiscompletes theproof.

The following lemma allows us to avoidexceptional sets E ⊂(r₀,R) the size of whichwith respect to theψ-measureisrestrictedinthesense thatψ-dens(E)<1.

Lemma 2.2. Let ψ ∈ D(r₀,R), and let f and g be non-decreasing functions defined on (r₀,R) satisfying f(r) ≤g(r) for all r ∈ (r₀,R)\E, where ψ-dens(E) < 1. Then, for any α >

1−ψ-dens(E)−1

, there exists anr ∈(r₀,R)such thatf(r)≤g(s(r))forallr∈(r,R),where s(r)=ψ⁻¹(αψ(r)).

Proof. Supposethere exists anincreasing sequence(r_n) on (r₀,R) tending to R suchthat[r_n,s(r_n)]⊂E foreveryn∈N.Define

I= ∞ n=1

[rn, s(rn)].

Then I⊂E, but

ψ-dens(I)≥lim sup

n→∞

1 ψ(s(rn))

I∩[r0,s(rn)]

dψ(t)

≥lim sup

n→∞

1 ψ(s(r_n))

[rn,s(rn)]

dψ(t) = 1− 1

α> ψ-dens(E),

which isacontradiction.Thus, there exists r > r0 such thatforany r≥r,there exists t ∈(r,s(r))\E.

Since f andgare non-decreasing,itfollowsthat

f(r)≤f(t)≤g(t)≤g(s(r)).

This completestheproof.

Lemma 2.2 generalizes [3, Lemma 3.1] and [9, Lemma 3.6], and also extends [16, Lemma 1.1.7]. In particular, thisversionworksforbothfinite andinfiniteintervals.

3. Upperandlowerlimits

Before consideringunboundedfunctionsT offinite orderor of finitelower order, weproceed to discuss upper and lower limits in general. The discussion thatfollows should be of independent interest, and it will be used for proving thegrowth results onthe functions T. More precisely, this section is devoted to provethefollowing result,whichisanimprovementof[3,Theorem 3.2] inthesensethatthepresentresult involves densitiesratherthanmeasures,anditworksforboth finiteandinfiniteintervals.

Theorem 3.1.Let0< r0< R≤+∞,and letϕ: (r0,R)→[0,∞)be afunctionwith lim sup

r→R⁻

ϕ(r) =K and lim inf

r→R⁻ ϕ(r) =k,

where 0≤k < K <∞.If thereexistsaψ∈ D(r0,R)suchthatϕ(r)ψ(r)isnon-decreasingon (r0,R),then forany ε>0,thesets

Fε={r∈(r0, R) :|ϕ(r)−K|< ε} and Gε={r∈(r0, R) :|ϕ(r)−k|< ε} satisfy

ψ-dens(F_ε)≥ ε

K, ψ-dens(G_ε)≥ ε

k+ε, (3.1)

ψ-dens(Fε)≤ k

k+ε, ψ-dens(Gε)≤ K−ε

K . (3.2)

In addition,

e^ψ-dens(Fε) =e^ψ-dens(Gε) = 1, e^ψ-dens(Fε) =e^ψ-dens(Gε) = 0.

If K= +∞,then foreverylarge M >0,thesetHM ={r∈(r0, R) :ϕ(r)> M}satisfies ψ-dens(H_M) = 1 and ψ-dens(H_M)≤ k

M. (3.3)

Proof. We prove the first inequality in (3.1).From the definition of lim sup, there exists an r1 ∈ (r0,R) suchthatϕ(r)< K+εforallr∈(r₁,R).HencethesetsF_εandF ={r∈(r₀, R) :ϕ(r)> K−ε}havethe sameψ-density.It sufficesto provenowthatψ-dens(F)≥ε/K. Assumethatψ-dens(F)< ε/K.Here,we mightassumethatε< K.

Letε andαsatisfy

0< ε< ε−K·ψ-dens(F)

2−ψ-dens(F) and α= K−ε K−ε+ε.

Then α >(1−ψ-dens(F))⁻¹. Wehaveϕ(r)≤K−εforr /∈F.Then ϕ(r)ψ(r)≤(K−ε)ψ(r) for r /∈F.

UsingLemma2.2withf(r)=ϕ(r)ψ(r) andg(r)= (K−ε)ψ(r) yields

ϕ(r)ψ(r)≤(K−ε)ψ(s(r)), r∈(r, R), (3.4) wheres(r)=ψ⁻¹(αψ(r)).Thus,

K= lim sup

r→R⁻

ϕ(r) = lim sup

r→R⁻

ϕ(r)ψ(r) ψ(r)

≤(K−ε) lim sup

r→R⁻

ψ(s(r))

ψ(r) = (K−ε)α < K−ε, whichisacontradiction.Henceψ-dens(Fε)=ψ-dens(F)≥ε/K.

Next,weprovethesecond inequalityin(3.1).From thedefinitionoflim inf,there existsanr₁∈(r₀,R) such that ϕ(r)> k−ε for all r ∈ (r₁,R). Hencethe sets G_ε and G = {r∈(r₀, R) :ϕ(r)< k+ε} have thesameψ-density.Therefore,it sufficesto provethatψ-dens(G)≥ε/(k+ε).Assumethatψ-dens(G)<

ε/(k+ε).Nowletε andαsatisfy

0< ε <ε−(k+ε)ψ-dens(G)

2−ψ-dens(G) and α=k+ε−ε k+ε .

Then α >(1−ψ-dens(G))⁻¹. Wehave (k+ε)ψ(r)≤ϕ(r)ψ(r) for r /∈ G. Using Lemma2.2 with f(r)= (k+ε)ψ(r) andg(r)=ϕ(r)ψ(r) yields

(k+ε)ψ(r)≤ϕ(s(r))ψ(s(r)), r∈(r, R), wheres(r)=ψ⁻¹(αψ(r)).Thus

k+ε≤lim inf

r→R⁻ ϕ(s(r)) lim sup

r→R⁻

ψ(s(r)) ψ(r)

=α lim inf

s(r)→R⁻ϕ(s(r)) =αk < k+ε−ε, whichisacontradiction.Henceψ-dens(G_ε)=ψ-dens(G)≥ε/(k+ε).

Toprovethefirstinequalityin(3.2),wefirstclaimthatthereexistsanr^∗∈(r₀,R) suchthatF_ε∩G_ε⊂ (r0,r^∗).Toprovethis claim,assumethecontrarythatthereexists anincreasingsequence(rn) onFε∩Gε

suchthatrn→R⁻ asn→ ∞. Then

|K−k| ≤ |f(rn)−k|+|f(rn)−K| →0, n→ ∞, and thisleadstoK=k,whichisacontradiction.Thustheclaimistrue.Therefore,

F_ε⊂G^c_ε∪(r₀, r^∗). (3.5)

Since ψ-dens((r0,r^∗))= 0,itfollows that

ψ-dens(F_ε)≤ψ-dens(G^c_ε) = 1−ψ-dens(G_ε)≤1− ε

k+ε = k k+ε. Similarly,wegetthesecond inequalityψ-dens(Gε)≤(K−ε)/K in(3.2).

Now, assume that e^ψ-dens(Fε) < 1. Set ψ1(r) = e^ψ(r). We then have ϕ(r) ≤ K−ε for r /∈ Fε and ψ1-dens(Fε)<1 byassumption.Therefore,(3.4) holdswiths(r)=ψ⁻¹₁ (αψ1(r)).Thus

K= lim sup

r→R⁻ ϕ(r) = lim sup

r→R⁻

ϕ(r)ψ(r) ψ(r)

≤(K−ε) lim sup

r→R⁻

ψ(s(r)) ψ(r)

= (K−ε) lim sup

r→R⁻

ψ(r) + logα

ψ(r) =K−ε,

which isacontradiction.Hencee^ψ-dens(Fε)= 1. Similarly,we canprovee^ψ-dens(Gε)= 1.The equalities e^ψ-dens(F_ε)=e^ψ-dens(G_ε)= 0 followfrom(3.5).

Finally,toprovetheinequalitiesin(3.3),weassumefirstthatψ-dens(HM)<1.Thenϕ(r)ψ(r)≤M ψ(r) for r /∈HM. ByLemma2.2,we obtainϕ(r)≤αM foreveryr nearR,which isacontradiction.Thusthe firstinequalityin(3.3) holds.Next,toprovethesecondinequalityin(3.3),weseethat,similarlytotheset Gabove,theset

H_M^c ={r∈(r₀, R) :ϕ(r)≤k+ (M−k)} satisfies ψ-dens(H_M^c )≥(M−k)/M.Thusψ-dens(HM)≤ k/M. Proposition 3.2. Foreveryε∈

0,^K−₂^k

,theinequalitiesin(3.1) and(3.2)canbe improved to ψ-dens(Fε)≥1−k+ε

K , ψ-dens(Gε)≥1− k

K−ε, (3.6)

ψ-dens(Fε)≤ k

K−ε, ψ-dens(Gε)≤ k+ε

K . (3.7)

Proof. Weproveonlythefirstinequalityin(3.6),andtherestoftheinequalitiesfollowsimilarly.Weusethe previousinequalitiesin(3.1) and(3.2).FromtheproofofTheorem3.1,weknowthatthesetsF_ε andF = {r∈(r0, R) :ϕ(r)> K−ε}havethesameψ-density,andthesetsGεandG={r∈(r0, R) :ϕ(r)< k+ε} havethesameψ-density.Then,from thesecondinequalityin(3.2),theset

F^c={r∈(r0, R) :ϕ(r)≤k+ (K−k−ε)} satisfies

ψ-dens(F^c)≤K−(K−k−ε)

K =k+ε

K .

Thenψ-dens(F_ε)=ψ-dens(F)≥1−(k+ε)/K.

Iftwofunctionsϕ1,ϕ2: [r0,R)→[0,∞) satisfyϕ1(r)< ϕ2(r),thenclearly lim sup

r→R⁻ ϕ1(r)≤lim sup

r→R⁻ ϕ2(r).

Conversely,if

lim sup

r→R⁻ ϕ1(r)<lim sup

r→R⁻ ϕ2(r),

then what can be said about the size of the set {r∈(r0, R) : ϕ1(r) < ϕ2(r)}? The next consequenceof Theorem3.1 givesthesize ofthissetbymeansoftheψ-density.

Corollary3.3. Letϕ1,ϕ2: (r0,R)→[0,∞)be functionsdefined on(r0,R) andsatisfying lim sup

r→R⁻

ϕ1(r) =k1< k2= lim sup

r→R⁻

ϕ2(r), (3.8)

and let ψ∈ D(r0,R) be such that ϕ2(r)ψ(r) is non-decreasing on (r0,R). Thenϕ1(r)< ϕ2(r) holds in a set G⊂(r0,R)with ψ-dens(G)≥1−k1/k2 and e^ψ-dens(G)= 1. The same conclusionshold if lim sup is replacedwith lim inf onboth sides of(3.8).

Proof. Supposefirstthatk2<∞.Let0< ε< k2−k1.Bythedefinitionoflim sup,

ϕ1(r)< k1+ε=k2−δ(ε) (3.9)

holdsforallr > r1> r0,where δ(ε)=k2−k1−ε>0.ByTheorem3.1,

ϕ₂(r)> k₂−δ(ε) (3.10)

holdsinaset G^∗with ψ-dens(G^∗)≥δ(ε)/k₂and e^ψ-dens(G^∗)= 1.From(3.9) and(3.10),theset G={r > r0:ϕ1(r)< ϕ2(r)}

satisfies ψ-dens(G) ≥δ(ε)/k2 and e^ψ-dens(G) = 1. Moreover, since G is independent on ε, it follows by lettingε →0⁺ that ψ-dens(G)≥(k2−k1)/k2. If k2 =∞, then similarly to the last partof theproof of Theorem3.1,wegetψ-dens(G)= 1.

Wecanprovesimilarlythatthesameconclusionsholdiflim sup isreplacedwithlim inf onbothsidesof (3.8).Thedetailsareomitted.

4. Growthofreal-valuedfunctions

The results inthis section are direct consequences of the results in Section 3, and they can easily be applied to obtain results on the growth of meromorphic functions. We restrict ourselves to study non- decreasingfunctionsontheinterval [1,∞).Fornon-decreasingfunctionson[0,1),analogous resultsfollow thesameway.

Corollary 4.1. Let T : [1,∞)→ (0,∞)be anon-decreasing unbounded functionof order L =ρ(T) and of lower order=ρ(T).If< a≤b< L,thenthesets

H={r≥1 :T(r)≤r^a} and I={r≥1 :T(r)> r^b} satisfy

logdens(H)≥max a−

a , L−a L+−a

, logdens(I)≤min

b,

L+−b

. (4.1)

logdens(H)≤min a

L,L+−a L

, logdens(I)≥max L−b

L ,b− L

. (4.2)

Proof. Toprovethefirstinequalityin(4.1),weapply Theorem3.1andProposition3.2with ϕ(r) = logT(r)

logr , ψ(r) = logr, ε=a−.

To provethesecond inequalityin(4.1),we noticethattheset I^c,which isthecomplementof I in[1,∞), satisfies

logdens(I^c)≥max b−

b , L−b L+−b

. Then, from(2.1),thesecond inequalityin(4.1) follows.

Theinequalitiesin(4.2) canbeprovedsimilarly.

Corollary 4.1 is an improvement of Theorem A. Moreover, the second inequality in (4.2) improves [6, Lemma 2.2], [8,Lemma 3] and [15,Lemma 2.7].

TwoparticularconsequencesofTheorem3.1canbestatedas follows.

Corollary 4.2. Let T : [1,∞) → (0,∞) be a non-decreasing function of order L ∈ (0,∞), and let ε > 0.

Then theset

K1= r≥1 :r^L−ε≤T(r)≤r^L+ε satisfies logdens(K1)≥ ε

L.

Corollary4.3. LetT : [1,∞)→(0,∞)beanon-decreasingfunctionoflowerorder∈(0,∞),andletε>0.

Then theset

K2= r≥1 :r^−ε≤T(r)≤r^+ε satisfies logdens(K2)≥ ε

+ε.

Corollary 4.2 improves [3, Corollary 3.3], which claims that the set K₁ has infinite logarithmic mea- sure. Replacing T(r) with T(r,f) for a meromorphic functionf, we see that Corollary 4.3 improves [15, Lemma 2.2],whichclaimsthattheset K2 hasinfinitelogarithmicmeasure.

AnotherconsequenceofTheorem3.1isstatedintermsof thetypeofgrowth.Recallthat τ(T) = lim sup

r→∞

T(r) r^ρ

is the type of T with respect to its order ρ = ρ(T) ∈ (0,∞). We give the following improvement of [3, Corollary 3.4],whichclaimsthatthesetN₁ definedbelowhasinfinitelinearmeasure.

Corollary 4.4.Let T : [1,∞) → (0,∞) be a non-decreasing function of order ρ ∈ (0,∞) and of type τ∈(0,∞),and letε0>0.Thentheset

N₁={r≥1 : (τ−ε₀)r^ρ≤T(r)≤(τ+ε₀)r^ρ} satisfiesdens(N₁)≥1−_τ−ε0

τ

1/ρ

.

TheconclusionofCorollary 4.4followsbyapplyingTheorem 3.1with ϕ(r) =T(r)^1/ρ

r , ψ(r) =r, ε=τ^1/ρ−(τ−ε0)^1/ρ.

Letf beanentirefunctionoforderρ∈(0,∞) andoftypeτ∈(0,∞) definedwithrespecttologM(r,f).

Letε>0.Then[14,Lemma 8] claimsthattheset

N2={r≥1 : (τ−ε)r^ρ≤logM(r, f)≤(τ+ε)r^ρ}

has infinite logarithmic measure. It follows from Lemma 2.1 that a set of finite logarithmic measure has zeroupperlineardensity.HenceweseethatCorollary4.4isanimprovementof[14,Lemma 8].

Tocomparethegrowthbetweentwo functions,westatethefollowing consequenceofCorollary 3.3.

Corollary 4.5. Let T₁,T₂ : [1,∞) →(0,∞) be non-decreasing and unbounded functions such that ξ(T₁) <

ξ(T2),whereξstandsforeithertheorder orthelowerorder,thesameorderonbothsides oftheinequality.

Letφ: [1,∞)→(0,∞)beanynon-decreasingfunctionsuchthatlogφ(r)=o(logr)asr→ ∞.Thenthe set P={r≥1 :φ(r)T₁(r)< T₂(r)}

satisfies

logdens(P)≥1−ξ(T₁)

ξ(T2) and dens(P) = 1.

TheconclusionofCorollary 4.5followsbyusingCorollary 3.3with ϕ₁(r) = logT₁(r) + logφ(r)

logr , ϕ₂(r) = logT₂(r)

logr and ψ(r) = logr.

Corollary4.5improves[5,Lemma 7].AspecialcaseofCorollary4.5isimplicitlyprovedin[7,p. 347] in thecaseρ(T₂)<∞.

A possible choicefor φin Corollary 4.5 is φ(r)= (logr)^β, where β > 0.If we replace T₁(r) and T₂(r) byT(r,f) andT(r,g),respectively,wheref andgaremeromorphicfunctions,andifφisunbounded,then Corollary 4.5statesinparticular, that

T(r, f) =o(T(r, g)), r∈P. (4.3)

Thusf isasmallfunctionofgrelativetothesetP.Recallthatinthecomplexfunctiontheory,ameromor- phicfunctionf issaidtobeasmallfunctionofanothermeromorphicfunctiong,ifT(r,f)=o(T(r,g)) for all routsideof asetof finitelinear measure(or sometimesoutsideof aset of finitelogarithmic measure).

Small functions appear frequently inthe theories of complex differential and functional equations, which in turn typically relyongrowth estimates for logarithmic derivativesand forlogarithmic differences. The formerestimatesareusuallyvalidoutsideofexceptionalsetsoffinitelinear/logarithmicmeasure,whilethe exceptionalsets inthelatterestimatesmaygo upto upperlogarithmicdensity< ε. Hence,inmostcases, the definition of small functions could be relaxed to (4.3), where the set P is required to have positive logarithmic upperdensity.

Next,wegivearesultaboutcomparingthegrowthoftwofunctionsinthecasewhentheyhavethesame order butdifferent types.

Corollary 4.6. Let T1,T2 : (1,∞)→(0,∞) be continuous, non-decreasing functionsboth having order ρ∈ (0,∞),andτ(T1)< τ(T2).LetC∈(1,τ(T2)/τ(T1)).Thentheset

Q={r≥1 :CT1(r)< T2(r)} satisfies

dens(Q)≥1−C^1/ρ

τ(T₁) τ(T2)

1/ρ

.

This followsbyusingCorollary3.3with ϕ1(r) = C^1/ρT1(r)^1/ρ

r , ϕ2(r) = T2(r)^1/ρ

r and ψ(r) =r.

In[4,Lemma 4],itisshownthatforameromorphicfunctionf oforderρ,andforconstantsC₁>1 and C₂>1,theset

U ={r:T(C₁r, f)≥C₂T(r, f)} (4.4) satisfies

logdens(U)≤ρlogC1

logC2

. (4.5)

If eitherρ= 0 or C₂^1/ρ ≥C₁, then theinequality (4.5) ismeaningful, and itgives information about size of the set U. In the oppositecase when ρ>0 and C₂^1/ρ < C₁, the quantity ^ρ_log^log_C^C1

2 is largerthan 1, and hencewe mayconcludenothingfrom (4.5).Inthiscase,theset U isexpectedto belarge,and itssizecan beestimateddirectlybymeansofCorollary3.3withanadditionalassumptiononthetypeoff.Infact,we havethefollowingresult.

Corollary 4.7. Let T : [1,∞) → (0,∞) be a non-decreasing function of order ρ ∈ (0,∞) and of type τ ∈(0,∞).LetC₁>1andC₂>1besuchthat C₂^1/ρ< C₁.Thentheset

V ={r:T(C₁r)≥C₂T(r)} satisfies

dens(V)≥1−C₂^1/ρ C1

.

Thisfollowsbyusing Corollary3.3with ϕ1(r) =C2T(r)^1/ρ

r , ϕ2(r) = T(C1r)^1/ρ

r , ψ(r) =r.

ReplacingT(r) withT(r,f) inCorollary4.7,where f isameromorphicfunctionoforderρ∈(0,∞) and oftypeτ∈(0,∞),we findthatthesetU in(4.4) islargeinthesensethatdens(U)≥1−C₂^1/ρ/C1. 5. Behaviorofintegrablefunctionsatinfinity

Recently, Niculescu and Popovici [11,12] have discussed necessary conditions for the integrability of real-valued functionsbased onaconcept oflimit inlineardensity. Wewill generalize this conceptfor the ψ-density, where ψ∈ D(r₀,R). We saythata functionf : (r₀,R) →R has alimitl ∈ R inψ-densityas r→R⁻ iftheset{r∈(r₀,R):|f(r)−l|≥ε}haszeroψ-density,wheneverε>0.Wedenote thislimit by

d_ψ- lim

r→R⁻f(r) =l.

Thevalue+∞(resp.−∞)iscalledthelimitoff inψ-densityas r→R⁻,andwedenote it d_ψ- lim

r→R⁻f(r) = +∞ (resp.− ∞),

iffor each M ∈R,the set {r∈(r₀,R):f(r)≤M}(resp.{r∈(r₀,R):f(r) ≥M}) haszero ψ-density.

Clearly,if lim

r→R⁻f(r)=l,thendψ- lim

r→R⁻f(r)=l forany ψ∈ D(r0,R).Theconverseisnottrueingeneral.

Forexample,thefunction

f(r) =

1, forr∈[n, n+ 1/2ⁿ], n∈N,

0, otherwise, (5.1)

doesnothavealimitas r→ ∞, butdr- lim

r→∞f(r)= 0.

Weprovethefollowingresultwhichgivesageneralnecessary conditionforintegrablefunctions.

Theorem5.1. Letf : [r0,∞)→R be alocallyintegrable functionon [r0,∞).If _∞

r0 f(t)dt<∞,then for any ψ∈ D(r₀,∞) wehave

r→∞lim 1 ψ(r)

r r0

ψ(t)f(t)dt= 0. (5.2)

Moreover, if_∞

r0 |f(t)|dt<∞,then

dψ- lim

r→∞

ψ(r)

ψ(r)f(r) = 0. (5.3)

Proof. Letε>0 andψ∈ D(r₀,∞).Thenthere existsanr₁> r₀ suchthatforeveryr > r₁,

r r1

f(t)dt <ε

3 and 1

ψ(r)

r1

r0

ψ(t)f(t)dt < ε

3. Therefore, foreveryr > r1,wehave

1

ψ(r) r r0

ψ(t)f(t)dt =

1

ψ(r)

⎛

⎝

r1

r0

ψ(t)f(t)dt+ r r1

ψ(t)

⎛

⎝ t r1

f(s)ds

⎞

⎠

dt

⎞

⎠

≤ 1

ψ(r)

r1

r0

ψ(t)f(t)dt +

r r1

f(s)ds +

1

ψ(r) r r1

ψ(t)

⎛

⎝ t r1

f(s)ds

⎞

⎠dt

<ε 3+ε

3+ψ(r)−ψ(r1) ψ(r)

ε 3< ε, whichresultsin(5.2).

Now,assumethat_∞

r0 |f(t)|dt<∞.Letε>0 andSε={r > r0: _ψ^ψ(r)_(r)|f(r)|> ε}.Then,byusing(5.2), we get

0≤ 1 ψ(r)

Sε∩[r0,r)

ψ(t)dt≤ 1 εψ(r)

r r0

ψ(t)|f(t)|dt→0, r→ ∞,

whichmeansψ-dens(S_ε)= 0 for everyε>0,andhenceweget(5.3).

ThefirstpartofTheorem5.1generalizes[10,Theorem 0.1],whilethesecondpartgeneralizes[11,Theo- rems3–4] and[12,Theorem2].Thefirstpartcanbeusedtoshowthedivergenceof_∞

r0 f(t)dtasfollows:If there existsaψ∈ D(r0,∞) suchthat(5.2) doesnothold,then_∞

r0 f(t)dtdiverges.Thefollowingexample shows thatthefirstpartof Theorem5.1isstrongerthan[10,Theorem 0.1].

Example 5.2.Considertheimproperintegral

I= ∞ 2

sin²t tlogtdt.

Wehave,byL’Hospital’srule,

r→∞lim 1 r r 2

sin²t logt dt= 0.

From [10,Theorem 0.1], weconclude nothingabout theintegral I. However,if wetake f(r)= _r^sin_log^2r_r and ψ(r)= logrin(5.2),wefind

r→∞lim 1 logr

r 2

sin²t

t dt= lim

r→∞

1

2 logr(logr+ Ci(2r)) = 1 2,

whereCi(r)=−_∞

r cost

t dtisthecosineintegral.Thus,accordingtoTheorem5.1,theimproperintegralI diverges.

Itisnaturaltoaskwhetherthelimitinψ-density(5.3) canbeimprovedtotheusuallimit.Surprisingly, Theorem 3.1plays akeyrole infindingasufficient conditiontoensurethat(5.3) isimprovedtotheusual limit.Infact,wehavethefollowingresult.

Theorem 5.3. Let f : [r₀,∞) → R be a function satisfying _∞

r0 |f(t)|dt < ∞. Suppose there exists ψ ∈ D(r0,∞)suchthat oneof thefollowingholds:

(i) ψ(r)²|f(r)|/ψ(r)is non-decreasingon(r1,∞) forsomer1≥r0, (ii) |f(r)|/ψ(r) isnon-increasingon (r₁,∞)forsomer₁≥r₀. Then

rlim→∞

ψ(r)

ψ(r)f(r) = 0.

The following lemma, which relies on Theorem 3.1, is needed to prove Theorem 5.3 in the case when (i) holds.

Lemma 5.4.Let 0< r0 < R≤+∞, andlet f : [r0,R)→[0,∞). Supposethat there exists aψ∈ D(r0,R) such that dψ- lim

r→R⁻f(r) = l. Then either lim

r→R⁻f(r) = l or the function f(r)ψ(r) is not non-decreasing on (r0,R).

Proof. We consider the case l ∈ [0,∞) only since the case l = ∞ follows similarly. Suppose that there existsaψ∈ D(r0,R) such thatdψ- lim

r→R⁻f(r)=l.Moreover,supposeonthecontrarytotheassertionthat f(r)ψ(r) isnon-decreasingon(r₀,R) andthatf(r) doesn’thavealimitasr→R⁻,thatis,

k= lim inf

r→R⁻ f(r)= lim sup

r→R⁻

f(r).

Letε>0,and let

Aε={r∈(r0, R) :|f(r)−k|< ε} and Lε={r∈(r0, R) :|f(r)−l| ≥ε}.

First,weprovethatl=k.Assumeonthecontrarythatl=k.Then,from Theorem3.1we obtainthat ψ-dens(Lε) =ψ-dens(A^c_ε) = 1−ψ-dens(Aε)>0,

whichisacontradictionwiththedefinitionoflimitsinψ-density.Thusl=k.

Next,we provethatAεand L^c_ε are disjoint for allr≥r^∗, where r^∗ ∈(r0,R).Assumeon thecontrary thatthereexistsanincreasingsequence (rn) onAε∩L^c_ε suchthatrn→R asn→ ∞.Then

|k−l| ≤ |f(rn)−l|+|f(rn)−k|<2ε, n→ ∞,

and this leads to k = l, which is a contradiction. Therefore A_ε ⊂ L_ε∪(r₀,r^∗). It follows from this, Theorem3.1 andthedefinitionoflimitsinψ-density,that

0< ψ-dens(A_ε)≤ψ-dens(L_ε) +ψ-dens((r₀, r^∗)) = 0,

which is a contradiction. Thus, either lim

r→R⁻f(r)=l orf(r)ψ(r) isnotnon-decreasingon(r₀,R).

Proof of Theorem5.3. (i)FromTheorem 5.1,wehave d_ψ- lim

r→∞

ψ(r)

ψ(r)|f(r)|= 0.

Since

ψ(r) ψ(r)|f(r)|

ψ(r) isnon-decreasing,weinferfromLemma5.4that lim

r→∞

ψ(r)

ψ(r)f(r)= 0.

(ii)Lets(r)=ψ⁻¹(¹₂ψ(r)).Wehaveforeveryr > ψ⁻¹(2ψ(r1)), ψ(r)

ψ(r)f(r)

= 2|f(r)| ψ(r)

r s(r)

ψ(t)dt≤2 r s(r)

|f(t)|

ψ(t)ψ(t)dt= 2 r s(r)

|f(t)|dt

from whichitfollowsthat lim

r→∞

ψ(r)

ψ(r)f(r)= 0.

Thefollowing exampleillustrates Theorem5.3.

Example 5.5.Thefunction

f(r) = 1

r(logr)^β, β >1,

is integrable on(e,∞).By taking ψ(r)=r inTheorem 5.3, we see thatboth conditions (i)and (ii) hold and hence lim

r→∞rf(r)= 0.

If wetakeψ(r)= (logr)^β inTheorem5.3, thenwe seethatboth conditions(i)and(ii)hold andhence

rlim→∞rlogrf(r)= 0.

If we take ψ(r) = log(log(r)) in Theorem 5.3, then the condition (i) does not hold. Meanwhile, the condition(ii)holds andhence lim

r→∞rlog(r)log(log(r))f(r)= 0.

Acknowledgments

Heittokangas wantsto thank theSchoolof Mathematical Sciences at the Fudan Universityfor its hos- pitality duringhis visitinAugust 2019.Latreuch was supportedbythe DirectorateGeneral forScientific Research andTechnologicalDevelopment (DGRSDT)ofAlgeria. WangwassupportedbytheNaturalSci- ence FoundationofChina (No. 11771090).

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