Vol. 40 No Journal of Jiangxi Normal University Natural Science Jul p q -φ. p q
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1 40 4 Vol 40 No Journal of Jiangxi Normal UniversityNatural Science Jul p q -φ 2 * Nevanlinna p q-φ 2 p q-φ p q-φ O A DOI /j cnki issn λ - p q /f φ= lim log p n - r f log q φr Nevanlinna φr E ~ 3 4 φr 0+! 0+! f p q-φ p q-φ iia bσ p q f + f 2 φ= max a b σ p q f f 2 φ= max a b σ p q f φ= lim log p Tr f log q φr 2 if μ p q f φ= lim log p Tr f log q φr log p nr /f λ p q f φ= lim 2 φr 0+! 0+ log! q φr = lim log p Nr /f log q φr f λ - log p n - r /f p qf φ= lim p log q-φ q φr = lim log p N r /f log q φr λ p q f φ= lim log p nr /f log q φr λ - p qf 3 φ= lim log p n - r /f log q φr φr 0+! 0+! μ p q f φ= lim f p q-φ λ p q /f φ= lim log p nr f log q φr +! 0+! 2 ilim log p r log q φr = 0ii +! me = E dt α > lim log q φαr log q φr= m l E = E dt /t p q-φ p q-φ 4-5 p q-φ p q f f 2 p q σ p q f φ= a σ p q f 2 φ= b iσ p q f + f 2 φ max a b σ p q f f 2 φ max a b iif σ p q f φ= lim log p Tr f log q φr = lim log p Tr f log q φr = lim log p + Mr f log q φr log p + Mr f log q φr BAB
2 p q-φ p λ p q /A φ< σ p q A φ f f 2 q-φ 2 λ p q /f i φ < μ p q f i 2 φ i = 2 F = f f 2 λ - p + qf φ = f + Af = 0 λ p + q F φ= σ p + q F φ σ p q A φ A B 8 p q-φ A λ - p qa A A 0 < ia= φ< σ p q A φf n <!λ - na< σ n A 0 f σ σ n A max λ - nf λ - p q A φ max λ - p qf φ λ - p q /f n /f φ f λ p q /f φ< μ p q f φ fδ! f> 0 f max λ n+ f λ n+ /f σ n A max λ p + q f φ λ p + q /f φ max λ - nf λ - n /f σ p q A φ max λ - p qf φ λ - p q /f φ B A 0 < ia= 4 A f f 2 n <! f f max λ p q f max λ n f λ n f 2 < σ n A Π iπ < n σ n Π < σ n A g g 2 g + A+ Πg = F = f f 2 E = g g 2 F E max i λ /F i λ /E< n max λ n /F λ n /E< σ n A max λ n f λ n f 2 σ n A 4 p q-φ C A σ p q A φ> 0 f σ p + q f φ= σ p q A φ D A σ p q A φ> 0 f f 2 2 F = f f 2 max λ p+ q f φ λ p+ q f 2 φ= λ p+ q F φ= σ p+ q F φ σ p q A φ σ p + q F φ < σ p q A φ f = c f + c 2 f 2 c c 2 0f λ p + q f φ= σ p q A φ 2 C 2 λ p q /A φ< σ p q A φf λ p q /f φ < μ p q f φ σ p + q f φ= σ p q A φ 2 φ λ p q f 2 φ< σ p q A φ Π σ p q Π φ < σ p q A φ g g 2 22 F = f f 2 E = g g 2 F E max λ p q /F φ λ p q /E φ< σ p q A φ max λ p q g φ λ p q g 2 φ σ p q A φ FrGr0+! ifr Gr r E E iir E 0 Fr GrE +! α > r 0 > 0r > r 0 Fr Gαr f σ p q f φ= σ μ p q f φ= μν f rf log p ν f r lim log q φr = σ lim log p ν f r log q φr = μ 3 9 D j k q j q k i > j i 0 i = q α > A E +! α Γ B > 0 = r f k A f j B Tαr f log α rlog Tαr f r f Γ = k E k j Γ k -j
3 4 p q-φ f σ p q f φ= σ <! Ug σ p q f φ= σ k ε > 0f= Ue g E 0+! σ p q f φ= max σ p q U φ σ p q e g φ r E mr f k /f= Οexp p - σ + εlog q φr σ p q U φ= lim log p Nr /f log q φr ε > f log U - exp p σ + εlog q φr e α +β α + β λ λ = r E me <! Tr f /f ' 3N r f+ 7N r /f+ 4N r /f + 9 f σ p q f φ= Sr f /f ' Sr f /f ' = οtr f /f ' r!r E me <! 6 A D D 2 f f 2 F = σ p q f φ= lim log p Tr f log q φr f= Ue g 23 8 p q-φ r n! n = + /nr n < r n+ lim log p Tr n f log q φr n = σ p q f φ r n! n n > n r r n + /nr n log q φr n log p Tr n f log q φ + /nr n log q φr n = log p Tr n f log q φ + /nr n log ptr f log q φr 3 E =! r n + /nr n 3 φr n = n ii lim log p Tr f log q φr lim r n! log p Tr n f log q φr n = σ p qf φ lim log p Tr f log q φr σ p q f φ φ= lim log p Tr f log q φrm l E = σ p q f! n = n + /nrn r n dt t =! n = n log + /n=! σ <! U Vg f= Ue g V σ p q f φ= max σ p q U φ σ p q V φ σ p q e g φ λ p q f φ = λ p q U φ = σ p q U φ λ p q /f φ= λ p q V φ= σ p q V φ f f 2 D F Tr F= ΟN - r /F+ Tr A+ log r r E me <! ε > 0 exp - exp p σ + εlog q φr f exp p + σ + εlog q φr 4 7 f σ p q f φ< = r E me <!! ε > 0 E ! r E 4 VU V f = r U exp p + σ + ε /3log q φr V exp p + σ + ε /3log q φr e g exp p + σ + ε /3log q φr U exp - exp p σ + ε /3log q φr r E me <! 8 V exp - exp p σ + ε /3log q φr r E 2 me 2 <! 9 σ p - q g φ= σ p q e g φ σ p q f φ= σe g e - g = r E E 2 r e g e - g exp - exp p σ + ε /3log q φr 0 E = E E 2 5~ f f= g dgd μ p q g φ = μ p q f φ = μ σ p q f φ = σ p q g φ<!σ p q d φ = ρ < μ f = r g = Mr g ν g r
4 g A D E +! = r 0 E D 2 f f 2 F = f f 2 C = Wf f 2 A F C f n f = ν n gr + ο n Ν 43 5 f= g d f n = gn d + n- g j j =0 d j j n d' j d d n jn d C j j j n C j j j n j + j + + nj n = n f n f = gn g + n- j = 0 g j g C j j j n j j n d' d j dn jn d 2 Hadamard f f= g dgd Wiman-Valiron df E +! = r 0 E σ p q d φ = λ p q /f φ < μ p q f φ = g = Mr g μ p q g φ σ p q g φ= σ p q f φ g j g = ν j gr + ο j = n f n ν = g r f ( ) n + ο + n- j = 0 j-n ν g r + ο C j j j n d' j j j n d d n jn d 4 σ p q d φ= ρ < με0 < 5ε < μ - ρ r Tα r d exp p ρ + εlog q φα r 5 F C = F'2 - C 2-2FF 4F 2 2 f λ p q /f φ < μ p q f φ σ = σ p q A φ 0 = r g = Mr g E +! = r 0 E f f= ν g r 2 + ο 9 9ε > 0 E 2 = r E 2 A exp p + σ + εlog q φr = r 0 E E 2 g = Mr g ν g r r 2 + ο= A < α < α 3 5 exp p + σ + εlog q φr E 2 +! = r 0 E 2 ν g r r exp p + σ + εlog q φr d m d B Tα r d k exp p ρ + 2εlog q φα r m = n 2 α < α r > r 0 < α r 0 6 ν g r α r exp p + σ + εlog q φα r 2 μ p q g φ= μ p q f φ= μ > ρ r ν g r> exp p μ - εlog q φr 7 σ p + q f φ= σ p + q g φ= 67 = r 0 E E 2 r!g = Mr g lim log p + ν g r log q φr σ + ε ε σ p + q f φ σ ν g r/ j-n d' /d j d n /d j n r /v g r n-j exp p ρ + 3εlog q φα r r n-j exp pρ + 3εlog q φα r exp p μ - 2εlog q φr r 2 > φr iii mr A= mr- f /f= Οlog rtr f 23 r E 3 me 3 <! λ p q /A φ < σ p q A φε0 < 2ε < σ - λ p q /A φ
5 4 p q-φ Nr A exp p λ p q /A φ+ εlog q φr 24 7 E 4 +! r E 4 Tr A exp p σ - εlog q φr 25 23~ 25 r E 4 \E 3 r exp max λ p q /f φ λ p q /f 2 φ+ εlog q φr 32 exp p σ - εlog q φr Οlog rtr f + exp p λ p q /A φ+ εlog q φr 26 exp p σ - εlog q φr Οlog rtr f 26 σ - 2ε σ p + q f φ ε σ σ p + q f φ σ p + q f φ= σ = σ p q A φ 2 σ = σ p q A φ σ p + q f φ= σ p + q f 2 φ= σ p q A φ= σ λ - p + qf φ λ p + q F φ σ p + q F φ max σ p + q f φ σ p + q f 2 φ = σ λ - p + qf φ= σ p + q F φ λ - p + qf φ < σ p + q F φ 4-5 F 2 = C ( 2 F' 2 - F ( F ) - 2 F - 4 A) 27 C 27 2Tr F Tr F /F+ 2Tr F' /F+ Tr A+ Ο 2mr F' /F+ mr F /F+ Nr F' /F+ Nr F /F+ Nr A+ mr A+ Ο= Nr A exp p λ p q /A φ+ εlog q φr 3 λ p q /f i φ< μ p q f i φ i = 2r N r F N r f + N r f 2 β λ - p + qf φ < β < σ p + q F φ r N r /F exp p + q β log q φr 33 29~ r E 5 r Tr F= Οexp p + β log q φr σ p + q F φ β < σ p + q F φ λ - p + qf φ= λ p + q F φ= σ p + q F φ 2 3 f σ p q A φ> 0f e α +β α + β λ 5 Tr f /f ' ΟN r f+ N r /f+ N r /f r E 6 me 6 <! N r / f N r /f+ N r /A Tr f /f ' ΟN r f+ N r /f+ N r /A r E 6 me 6 <! 34 σ p q A φ > max λ - p qf φ λ - p q /f φ λ - p qa φ < σ p q A φ 34 σ p q f /f ' φ max λ - p qf φ λ - p q /f φ λ - p qa φ< σ p q A φ 35 Οmr F' /F+ mr F /F+ N r /F+ N r η = f ' /f 35 F+ Nr A+ mr A 28 σ p q η φ< σ p q A φ p q-φ r mr A Tr A exp p σ + εlog q φr 29 Tr A= Tr- η' + η 2 Tr η'+ 2Tr η+ Ο σ p q A φ max σ p q η φ σ p q η' φ= 4 E σ 5 p q η φ< σ p q A φ r E 5 mr F' /F= Οexp p σ + εlog q φr mr F /F= Οexp p σ + εlog q φr 30 σ p q A φ max λ - p qf φ λ - p q /f φ λ p q /f φ < μ p q f φ σ p + q f φ σ p q A φ λ p q /A φ< σ p q A φ= σr max λ p + q f φ λ p + q /f φ σ p q A φ max λ - p qf φ λ - p q /f φ 3 4 E = g g 2 λ p q E φ max λ p q g φ λ p q g 2 φ
6 λ p q E φ σ p q A φ λ p q E φ < σ p q A λ p q F φ max λ p q f φ λ p q f 2 φ < σ p q A φε > 0 r N r /F exp p λ p q F φ+ εlog q φr 36 Tr A exp p σ p q A φ+ εlog q φr 37 6 Tr F= ΟN r /F+ Tr A+ log r38 r E 7 me 7 <! 36~ 38 Tr F= Οexp p σ p q A φ+ εlog q φr+ log r r E 7 me 7 <! iii σ p q F φ σ p q A φ+ ε ε σ p q F φ σ p q A φ 39 4A = F'2 - C 2-2FF F 2 φ 44~ 46 9 φr Tr A + Π + C2 2 4 F C 40 C = Wf f Tr A 2Tr F' /F+ Tr F /F+ 2Tr /F+ Ο ΟTr F+ Tr F'+ Tr F 4 4 σ p q F φ σ p q A φ 39 σ p + q f φ= σ p q A φ σ p q R φ= λ p q R φ= λ p q E φ< σ p q A φ 45 max λ p q /F φ λ p q /E φ= max σ p q U φ σ p q V φ< σ p q A φ 46 σ p q F φ= σ p q E φ= σ p q A φ σ p q e P φ= σ p q e S φ= σ p q A φ Ce 2P -S = - U 2 R 2 V 2 Q 2 C 0 F 2 = V2 Q 2 E 2 U 2 R 2 e2p -S = - C A + Π + C2 2 C 2 C 2 2 C 2 2 C F F C A= C 2 Nr A + Π + C2 2 C 2 ( E' 2 E E ) - 2 E + C2 2 C 2 C A= mr C A + Π + C2 2 C 2 C A= F' ( F ) 2 - C A+ Οmr F' /F+ mr F /F+ mr E' /E+ mr E /E + N r /F + N r F + N r /E + N r E 48 α = max λ p q F φ λ p q E φ λ p q /F φ λ p q /E φ 44~ 46 α < σ p q A φ 4 48 E 8 0+! r E 8 4A + Π= E'2 - C 2 2-2EE 42 E 2 Tr A + Π + C2 2 C 2 C A= Οexp pα log q φr ε σ p q E φ= σ p q A + Π φ= σ p q A φ C 2 = Wg g 2 0 σ p q A + Π + C2 2 C 2 C A p qa φ 9 F E C = - C 2 F = Qe P /U E = Re S 2 /C 2 F 2 = C 2 E 2 /C 2 2 F' /F = E' /E /V 43 F /F = E /E 4042 Π = 0 Q U R V P S λ p q Q φ= σ p q Q φ= λ p q F φ< σ p q A φ 44 max λ p q g φ λ p q g 2 φ σ p q A φ 4 3 Hayman W K Meromorphic functions M OxfordClarendon Press 964 2Laine I Nevanlinna theory and complex differential equa-
7 4 p q-φ tions M BerlinWalter de Gruyter 993 3Hayman W K The local growth of power seriesa survey of the Wiman-Valiron method J Canad Math Bull Shen Xia Tu Jin Xu Hongyan Complex oscillation of a second-order linear differential equation with entire coefficients of p q-φ order J Adv Difference Equ φrp q J p q-φr J J Cao Tingbin Li Leimin Oscillation results on meromorphic solutions of second order differential equations in the complex plane J Electron J Qual Theory Differ Equ Gundersen G G Estimates for the logarithmic derivative of a meromorphic function plus similar estimates J J London Math Soc Hayman W K Picard values of meromorphic functions and their derivatives J Ann Math Bank S Laine I On the eros of meromorphic solutions of second-order linear differential equations J Comment Math Helv Tu Jin Long Teng Oscillation of complex high order linear differential equations with coefficients of finite iterated order J Electron J Qual Theory Differ Equ Jank G Volkmann L Untersuchungen ganer und meromorpher funktionen unendlicher ordnung J Arch Math Tu Jin Chen Zongxuan Growth of solutions of complex dif- ferential equations with coefficients of finite iterated order J Southeast Asian Bull Math Kinnunen L Linear differential equations with solutions of finite iterated order J Southeast Asian Bull Math Bank S Laine I Langley J K Oscillation results for solutions of linear differential equations in the complex domain J Results Math The Complex Oscillation of a Second Order Linear Differential Equation with Meromorphic Coefficients of p q-φ Order LUO Liqin ZHENG Xiumin * Institute of Mathematics and Informatics Jiangxi Normal University Nanchang Jiangxi China AbstractProperties of meromorphic solutions of a second order linear differential equation with meromorphic coefficients of p q-φ order are investigated by using Nevanlinna's value distribution theory of meromorphic functions And some results on the relations between the order of meromorphic solutions the convergence exponent of distincteros and distinctpoles of meromorphic solutions and the order of the coefficients are obtained which are improvements and extensions of the corresponding results of previous papers Key wordslinear differential equationmeromorphic coefficient p q-φ order p q-φ convergence exponent
Vol. 38 No Journal of Jiangxi Normal University Natural Science Nov. 2014
38 6 Vol 38 No 6 204 Journal o Jiangxi Normal UniversityNatural Science Nov 204 000-586220406-055-06 2 * 330022 Nevanlinna 2 2 2 O 74 52 0 B j z 0j = 0 φz 0 0 λ - φ= C j z 0j = 0 ab 0 arg a arg b a = cb0
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