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 < c < φz 0 0 λ - φ= λ ' - φ= λ - φ= 2 Nevanlinna + e az ' + B 0 e bz + B e dz = 0 2-2 σ z λ 8 z λ z D z 0 B i z 0i = 0 λ - φ z φ σ < σb i < a b d σ 2 z abd 0a max b d a - b a - λ 2 - φ z φ 2 d 0b = d B 0 z + B z 0 φz 0 2 0 2 λ - φ= λ ' - φ= λ - φ= 2 + e az ' + 0 e bz = 0 7 2 0 λ - j z 0j = 0 φ= λ ' - φ= λ - φ= σ= 0 2 3 2 2 j z 0j = 0 φz 0 0 λ 2 - φ = σ 2 φz 0 0 λ ' - φ= λ - φ= λ 2 ' - φ= λ 2 - φ= σ 2 2 j z 0j = 0 204-02-25 770 955-
552 204 φz 0 + e az ' + 0 e bz = F 3 Fz 0 λ 2 = λ 2 = σ 2 0 λ - φ = λ 2 - φ= σ 2 φz 0 Fz0 3 0 λ ' - φ = λ - φ= λ 2 ' - φ= λ 2 - φ= σ 2 3 j z 0j = 0 ab 0 arg a arg b a = cb0 < c < φz 0 0 0 λ - φ = λ ' - φ = λ - φ= λ 2 - φ= λ 2 ' - φ = h < k λ n r r E E 2 -φ= σ 2 j z 0j = 2 n 9 a j j = 0 k - F0 k + a k - k - + + a 0 = F 4 maxσf σa j j = 0 k - < σ= σ0 < σ λ= λ= σ a j j = 0 k - F 2 σ = 4 = F k k - + a k - + + a 0 5 5z 0 k α z 0 F α - k nr / knr /+ nr /F Nr / knr /+ Nr /F+ O r E mr j /= OlogrTr j = k mr / mr /F+ k - mr a j + j = 0 OlogrTr r E λ 2 - φ= λ 2 ' - φ= λ 2 - φ= σ 2 4 z 0B i z 0i = 0 Tr = Tr / + O knr / + σ < σb i < a b d abd 0a max b d a - b a - Tr /F + OlogrTr + k - Tr a j = j = 0 d 0b = d B 0 z + B z 0 knr /+ Tr F+ OlogrTr + φz 0 2 k - Tr j = 0 0 λ 2 - φ = λ 2 ' - φ = λ 2 - φ = σ 2 a j r E maxσf σa j j = 0 k - < σ= σ 2 λ 2 5 z 0B i z 0i = 0 λ 2 = λ 2 = σ 2 σ < σb i < a b d 3 0 z 2 z n z n 2 abd 0a max b d a - ba - g z g 2 z g n z d 0b = d B 0 z + B z 0 φz 0Fz 0 i n j ze g j z 0 j = + e az ' + B 0 e bz + B e dz = F iig j z- g k z j < k n iiitr j = otr e g h -g k j n 4 4 j z 0j = 0 F 0 3 0 5 3 a b ab 0 arg a arg b a = cb0 < c < Λ = 0 a Λ 2 = 0 a b a + b 2a i H j j Λ H b H b z 0ψ z = j Λ H j ze jz
6 2 553 ψ z+ H b e bz 0 + e az - 0e bz ' + iih j j Λ 2 H 2b 0 e bz e az ' + H 2b z 0ψ 2 z = H j ze jz j Λ 0 e bz - 0e bz ' 2 0 e bz e az ' = 0 9 ψ 2 z+ H 2b e 2bz 0 ' = g + φ = g' + φ' = g + iii ψ 20 z ψ 2 z ψ 22 z ii 9 ψ 2 z H 2b z 0 φz g + h g' + h 0 g = h 0 h = e az - 0 e bz ' 0 e bz h 0 = e az ' + z φz ψ 22z+ φ'z φz ψ 2z+ ψ 20 z+ H 2b e 2bz 0 6 z B i z i = 0 σ< σb i < a max b d 2 0 2 6 z0 B i z 0i = 0 abd 0 2 0 σ= g 0 z= z- φz σg 0 = σ= B λ - φ = λg 0 = λ 2 - φ = σg 2 = σ = σ= λ 2 - φ= λ 2 g 0 = g 0 + φ λ g 0 + e az g' 0 + 0 e bz 2 g 2 7 g 0 = 4 + - + e az φ' + 0 e bz e az + 2 e az ' + 0 e bz + φ 6 e az + 2 0 e bz ' ' + 0 e bz = 0 6 89 g 0 = - φ 6 4 + H 3 + H 2 = 0 2 g 0 λ 2 g 0 = σ 2 φz H 3 = e az - U /U 2 H 2 = 2 e az ' + X 0 = + e az φ' + 0 e bz φ 0 σx 0 < σg 0 = σ 0 e az = σ e bz = < σg 0 = 6 2 λ 2 g 0 = λ 2 g 0 = σ 2 g 0 λ 2 - φ= σ 2 λ 2 ' - φ= σ 2 g = ' - φσg = σ '= σ= λ 2 - φ= λ 2 g + e az + e az ' + 0 e bz ' + 0 e bz ' = 0 7 = - 0 e bz + e az ' 8 87 0 e bz - 0 e bz ' e az 0 e bz - h = + e az - 0 e bz ' 0 e bz φ' + e az ' + 0 e bz - 0 e bz ' e az 0 e bz φ = - ' 0 / 0 + bφ' + φ' + ' φ + aφ - ' 0 / 0 + b φ e az + 0 φe bz h 0 h 0 φ - ' 0 0 + b φ' φ + φ' φ + ' + a - ' 0 + 0 b e az + 0 e bz = 0 3 0 0 h 0 λg = λg = σg = λ ' - φ = 2 λ 2 g = λ 2 g = σ 2 g λ 2 ' - φ= σ 2 λ 2 - φ= σ 2 g 2 = - φ 0 e bz - 0e bz - U 0 e bz U e az - 0e bz ' 2 0 e bz U = e az + 2 0 e bz ' - e az 0 e bz 0 e bz U 2 = e az ' + 0 e bz - e az 0 e bz ' 0 e bz H 3 H 2 U U 2 = g 2 + φ = g' 2 + φ' 4 = g 2 + 2 g 2 + H 3 g' 2 + H 2 g 2 = - + H 3 φ' + H 2 φ 3 X = + H 3 φ' + H 2 φ X 0 λg 2 = σg 2 = λ - φ= 2 λ 2 g 2 = σ 2 g 2 λ 2 - φ= σ 2 2 0 3
554 204 4 φ' 0 σ = g 3 z = φ + 0' + 0 a - ' 0 + 0 b e a+bz + z- φz σφ< σg 3 = σ= 2 0e 2bz - F' φ 0e bz + F φ = 0 λ - φ= λg 3 λ 2 - φ= λ 2 g 3 σ j < j = 0 σf< σφ< = g 3 + φ 3 3 2 g 3 + e az g' 3 + 0 e bz 0 0 0 0 g 3 = h F - + e az φ' + 0 e bz 2 0 φ 4 λ ' - φ = λg X = F - + e az φ' + 0 e bz 4 = φ= F - X 0 σg X 0 4 = σ = 2 λ 2 g 4 = φ 3 σ 4 φz 2 g 4 λ 2 ' - φ= σ 2 φ φ 4 λg 3 = λg 3 = g 5 λ - φ= λ 2 - φ= σ 2 = - φσg 5 = σ = σ = σg 3 = λ - φ= λ - φ= λg 5 λ 2 - φ= λ 2 g 5 5 2 λ 2 g 3 = σ 2 g 3 4 + λ 2 - φ= σ 2 e az + 2 e az ' + 0 e bz + e az + 2 0 e bz ' ' + 0 e bz = F 8 687 λ ' - φ= λ 2 ' - φ= σ 2 4 + H g 4 = ' - φσg 4 = σ ' = σ = 3 + H 2 = H 9 λ ' - φ= λg 4 λ 2 ' - φ= λ 2 g 4 H 3 = e az - U /U 2 H 2 = 2 e az ' + 3 + e az + e az ' + 0 e bz ' + 0 e bz - 0e bz - U 0 e bz 0 e bz U e az - 0e bz ' 2 0 e bz ' = F' 5 3 H = F - U F ' - F 0e bz = - U 2 0 e bz- F 0 e bz 0 e bz + e az ' - F 6 U = e az + 2 0 e bz ' - e 0e bz az 6 5 0 e bz + e az - 0 e bz ' 0 e bz + e az ' + U 2 = e az ' + 0 e bz - e 0e bz ' az 0 e bz - 0e bz ' 0 e bz e az ' = F ' - F 0 e bz 0 e bz 7 H 3 H 2 H U U 2 ' = g 4 + φ = g' 4 + φ' = g 4 + 7g 4 + h g' 4 + h 0 g 4 = h = g 2 5 + φ = g' 5 + φ' 4 = g 5 + 9 h = e az - 0 e bz ' 0 e bz g h 0 = e az ' + 0 e bz - 0 e bz ' e az 0 e bz 5 + H 3 g' 5 + H 2 g 5 = H - + H 3 φ' + H 2 φ X 2 = H - + H 3 φ' + H 2 φ - h 2 = - ' 0 / 0 + bφ' + φ' + ' φ + X aφ - ' 0 / 0 + b φ e az + 0 φe bz 2 0 - F' + λ - φ= λg F / 0 e bz 5 = σg 5 = σ= 2 λ 2 g 5 = σ 2 g 5 h 2 0 h 2 0 φ - ' 0 0 + b φ' φ + φ' φ + ' + a - ' 0 + b e az + 0 e bz - F' 0 φ + F 0 e bz φ = 0 0 e bz 0 φ - ' 0 + 0 b φ' φ - F' φ 0 e bz + λ 2 - φ= σ 2 2 λ - φ= λ ' - φ= λ - φ = λ 2 - φ = λ 2 ' - φ= λ 2 - φ= σ 2 3 0 σ= C
6 2 555 λ - φ= λ ' - φ= λ - φ= = - 3 B 0 e bz + B e dz + eaz ' 22 λ 2 - φ= σ 2 222 λ 2 ' - φ= σ 2 g = ' - φ + e az - B 0e bz + B e dz ' + e az ' + 7~ 0 B 0 e bz + B e dz h 0 h 0 B 0 e bz + B e dz - B 0e bz + B e dz ' e az ' = 0 23 /φ - ' 0 / 0 + bφ' /φ + φ' /φ + ' + B 0 e bz + B e dz a - ' 0 / 0 + b e az + 0 e bz ' = g = 0 + φ = g' + φ' = g + 23g 5 iii + h g' + h 0 g = h h /φ - ' 0 / 0 + bφ' /φ + φ' /φ + ' + = e az - B 0 e bz + B e dz ' B 0 e bz + B e dz a - ' 0 / 0 + b e az + 0 e bz h 0 0 = e az ' + B 0 e bz + B e dz - B 0 e bz + B e dz 'e az h 0 B 0 e bz + B e dz 2 λ 2 g = λ 2 g = - h = + e az - B' 0 + B 0 be bz + B' + σ 2 g λ 2 ' - φ= σ 2 B de dz ' B 0 e bz + B e dz φ' + ' + ae az + λ 2 - φ= σ 2 g 2 ~ 3 = - φ X = + H 3 φ' + H 2 φ h 0 h 0 X 0 B 0 φ - B' 0 + B 0 b φ' φ ebz + B φ - B' + 2 λ 2 g 2 = σ 2 g 2 λ 2 - φ= σ 2 B d φ' φ φ' edz + B 0 φ - B' 0 + B 0 b+ B 0 ' + 4 0 2 6 σ= g 0 z= z- φz σg 0 = σ= D λ - φ= λ ' - φ= λ - φ= g 0 = - φ 20 g 0 λ 2 g 0 = σ 2 φz 0 X 0 = + e az φ' + B 0 e bz + B e dz φ 0 2 λ 2 g 0 = σ 2 g 0 λ 2 - φ= σ 2 λ 2 ' - φ= σ 2 g = ' - φ σg = σ ' = σ = λ 2 ' - φ = λ 2 g 2 + e az + e az ' + B 0 e bz + B e dz ' + B 0 e bz + B e dz ' = 0 2 2 B 0 e bz + B e dz - B' 0 + B 0 be bz + B' + B de dz 'e az B 0 e bz + B e dz φ a e a+bz φ' + B φ - B' + B d+ B ' + a e a+dz + B 2 0e 2bz + B 2 e 2dz + 2B 0 B e b+dz = 0 24 2 λ 2 - φ= λ 2 g 0 b = d a max b = g 0 + φ 2 d a b24 g 0 + e az g' 0 + B 0 e bz + B e dz g 0 = B 0 + B - + e az φ' + B 0 e bz + B e dz φ - B' 0 + B' + B 0 + B 2 b φ' φ ebz + φ 20 20 B 0 + B φ' φ - B' 0 + B' + B 0 + B b+ B 0 +B ' + a e a+bz + B 0 + B 2 e 2bz = 0 B 0 B φ 3 2 B 0 + B 2 0 2 b d a - ba - d 0 a b a d24 3 2B 0 B 0 2 h 0 2 λ 2 g = σ 2 g λ 2 ' - φ= σ 2 λ 2 - φ= σ 2 g 2 = - φσg 2 = σ = σ= λ 2 - φ= λ 2 g 2 2 4 + e az + 2 e az ' + B 0 e bz + B e dz + e az + 2B 0 e bz +
556 204 B e dz ' ' + B 0 e bz + B e dz = 0 25 222523 4 + H 3 + H 2 = 0 26 H 3 = e az - U /U 2 H 2 = 2e az ' + B 0 e bz + B e dz - B 0e bz + B e dz - U h B 0 e bz + B e dz U U = e az + 2 2B 0 e bz + B e dz ' - e B 0e bz + B az e dz J 2002 B 0 e bz + B e dz 262-27 U 2 = e az ' + B 0 e bz + B e dz - 6 e B 0e bz + B az e dz ' J 2003 272 B 0 e bz + B e dz 8-2 7 + e - z ' + Qz = 0 H 3 H 2 U U 2 J 200 39 775-785 = g 2 + φ = g' 2 + φ' 4 = g 2 + 26 g 2 + H 3 g 2 ' + H 2 g 2 = - + H 3 φ' + H 2 φ X = + H 3 φ' + H 2 φ 0 2 λ 2 g 2 = σ 2 g 2 λ 2 - φ= σ 2 3 Hayman W K Meromorphic unction M OxordClarendon Press 964 2 M 982 3 J 2006 27443-442 4 J 2002 265-20 5 8 2 J 203 37 3 233-235 9Chen Zongxuan Zeros o meromorphic solutions o higher order linear dierential equations J nalysis 994 4 4 425-438 0Yang Congjun Yi Hongxun Uniqueness theory o meromorphic unctions M New YorkKluwer cademic 5 6 2 Publishers 2003 3 4 5 Cheng Tao Kang Yueming The growth o the solutions o a certain linear dierential equation J Journal o Fudan University 2006 455 6-68 The Relations between Solutions o Second Order Linear Dierential Equations with Functions o Small Growth MIN Xiao-hua ZHNG Hong-xia YI Cai-eng * College o Mathematic and Inormatics Jiangxi Normal University Nanchang Jiangxi 330022 China bstractit was investigated that the relations between solutions o second order linear dierential equations and their th and 2 th derivatives with the small growth unctions by using the theory and the method o Nevanlinna value distribution The precision result was obtained that convergence exponents o various points o equation solutions and their derivatives etch the small growth unction is ininite and the 2th convergence exponents with the hyper order o solution is equal Key wordsdierential equationentire unctionhyper-order2th exponents o convergence