40 4 Vol 40 No 4 206 7 Journal of Jiangxi Normal UniversityNatural Science Jul 206 000-586220604-033-07 p q -φ 2 * 330022 Nevanlinna p q-φ 2 p q-φ p q-φ O 74 52 A DOI0 6357 /j cnki issn000-5862 206 04 0 0 λ - p q /f φ= lim log p n - r f log q φr Nevanlinna φr 0-2 3 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 206-03-2 30233 205BAB20004 980-
332 206 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 3 4 3 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 2 2 2 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 = 0 2 2 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 2 2 4 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
4 p q-φ 2 333 4 4 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 > 0 5 0 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 =! 7 8 2 4 3 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 + 23 7! 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 5 6 7 8 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~ 0 4 9 0 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
334 206 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 3 2 3 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 20 920 = 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 0 8 8 4 0 r 2 > 0 22 22 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
4 p q-φ 2 335 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~ 33 28 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
336 206 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 2 0 40 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 43 40 6 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 47 40 4247 4A + 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-
4 p q-φ 2 337 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 974 7337-358 4Shen 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 204 200-4 5 φrp q J 202 36 47-50 6 p q-φr J 205 392207-24 7 J 203 37 5449-452 8Cao Tingbin Li Leimin Oscillation results on meromorphic solutions of second order differential equations in the complex plane J Electron J Qual Theory Differ Equ 200 68-3 9Gundersen G G Estimates for the logarithmic derivative of a meromorphic function plus similar estimates J J London Math Soc 988 372 88-04 0Hayman W K Picard values of meromorphic functions and their derivatives J Ann Math 959 70 9-42 Bank S Laine I On the eros of meromorphic solutions of second-order linear differential equations J Comment Math Helv 983 58656-677 2Tu Jin Long Teng Oscillation of complex high order linear differential equations with coefficients of finite iterated order J Electron J Qual Theory Differ Equ 200966-3 3Jank G Volkmann L Untersuchungen ganer und meromorpher funktionen unendlicher ordnung J Arch Math 982 3932-45 4Tu Jin Chen Zongxuan Growth of solutions of complex dif- ferential equations with coefficients of finite iterated order J Southeast Asian Bull Math 2009 3353-64 5Kinnunen L Linear differential equations with solutions of finite iterated order J Southeast Asian Bull Math 998 224 385-405 6Bank S Laine I Langley J K Oscillation results for solutions of linear differential equations in the complex domain J Results Math 989 63-5 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 330022 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