Математичнi Студiї. Т.4, 2 Maemaychni Sudii. V.4, No.2 УДК 57.5 K. G. Malyuin, T. I. Malyuina, I. I. Kozlova ON SUBHARMONIC FUNCTIONS IN THE HALF-PLANE OF INFINITE ORDER WITH RADIALLY DISTRIBUTED MEASURE K. G. Malyuin, T. I. Malyuina, I. I. Kozlova. On subhamonic funcions in he half-plane of infinie ode wih adially disibued measue, Ma. Sud. 4 (24), 79 83. If a pope subhamonic funcion of infinie ode has he full measue a he finie sysem of ays in he uppe half-plane, hen is lowe ode also equals infiniy. К. Г. Малютин, Т. И. Малютин, И. И. Козлова. Субгармонические функции в полуплоскости бесконечного порядка с радиально распределенной мерой // Мат. Студiї. 24. Т.4, 2. C.79 83. Если полная мера истинно субгармонической функции бесконечного порядка распределена на конечной системе лучей в верхней полуплоскости, то ее нижний порядок также равен бесконечности.. In his pape we use he Fouie seies mehod o sudy he popeies of subhamonic funcions. This mehod was inoduced by L. A. Rubel and B. A. Taylo ([]). Fuhe he Fouie seies mehod was used by J. B. Miles ([2]), A. A. Kondayuk ([3, 4, 5]) and ohes. Le v be a subhamonic funcion in he complex plane C, M(v, ) = max θ 2 v(e iθ ). The ode and lowe ode of he funcion v ae defined o be he values ln M(v, ) β[γ] = lim ln, α[γ] = lim ln M(v, ). ln The ode and lowe ode of an enie funcion f ae defined as he ode and lowe ode of he subhamonic funcion ln f, especively. In [6] he auho consideed he enie funcions which zeos lie on he finie sysem of ays. In paicula, i was poved ha if f is an enie funcion of infinie ode wih posiive zeos hen is lowe ode equals infiniy as well. This esul is easily genealized o he subhamonic funcions in he complex plane: if he Riesz measue of a subhamonic funcion in he enie complex plane v of infinie ode is locaed on a posiive half-axis hen is lowe ode also equals infiniy. We pove a simila esul fo funcions which ae subhamonic in he half-plane. The special case, whee he measue is disibued on he imaginay axis, was consideed in [7]. 2. Le C + = {z : Im z > } be he uppe half-plane of he complex vaiable z. We denoe by C(a, ) he open disc of adius wih cene a a, and by Ω + he inesecion of a se Ω wih he half-plane C + : Ω + = Ω C + ; G means closue of a se G. If < < 2 hen D + (, 2 ) = C + (, 2 )\C + (, ) means a closed half-ing. 2 Mahemaics Subjec Classificaion: 3A5, 3A. Keywods: pope subhamonic funcion; infinie ode; lowe ode; Fouie coefficiens; full measue. c K. G. Malyuin, T. I. Malyuina, I. I. Kozlova, 24
8 K. G. MALYUTIN, T. I. MALYUTINA, I. I. KOZLOVA Le SK be he class of subhamonic funcions in C + possessing a posiive hamonic majoan in each bounded subdomain of C +. Funcions v(z) fom SK have he following popeies ([8]): a) v(z) has non-angenial limis v() almos eveywhee on he eal axis and v() L loc (, ); b) hee exiss a measue of vaiable sign ν on he eal axis such ha b lim y + a v( + iy)d = ν([a, b]) 2 ν({a}) 2 ν({b}). The measue ν is called he bounday measue of v; c) dν() = v()d + dσ(), whee σ is a singula measue wih espec o he Lebesgue measue. Fo a funcion v SK, following [8], we define he full measue λ by seing λ(k) = 2 Im ζdµ(ζ) ν(k), C + K whee µ is he Riesz measue of v. A subhamonic in C + funcion v is said o be pope subhamonic if lim sup z v(z) fo all eal numbes R. Denoe he class of pope subhamonic funcions by JS. The full measue of he funcion v JS is a posiive measue, which explains he em pope subhamonic funcion. The class of dela-subhamonic funcions Jδ is defined o be he diffeence Jδ = JS JS. Fo a funcion v Jδ he epesenaion in a disc z C + (, R) is well defined v(z) = 2 C + (,R) G(z, ζ) Im ζ dλ(ζ) + R 2 G(z, Re iϕ ) v(re iϕ )dϕ, () τ whee G(z, ζ) is he Geen funcion of he half-disc, G means he deivaive in he inwad τ nomal diecion, and he kenel of double inegal is exended by coninuiy o he eal axis fo R. Fo he measue λ denoe λ() = λ(c(, )). Le v Jδ, v = v + v, λ be he full measue of v, λ = λ + λ be he Jodan decomposiion of measue λ. Le us inoduce he following chaaceisics of he funcion v m(, v) := v + (e iϕ ) sin ϕdϕ, N(, v, ) := λ () d, 3 T (, v, ) := m(, v) + N(, v, ) + m(, v), >, whee is an abiay fixed posiive numbe (one may as well ake = ) which in designaions (if i does no cause a misundesanding) we will no wie (fo example, insead of T (, v, ) will wie T (, v) and so on). Le λ k () = λ k (C(, )) whee dλ k (τe iϕ sin kϕ ) = τ k dλ(τe iϕ ), k N (he funcion sin ϕ sin kϕ/ sin ϕ is defined fo ϕ =,, by coninuiy).
ON SUBHARMONIC FUNCTIONS IN THE HALF-PLANE 8 Noe he Caleman s fomula in Gishin s noaion k In paicula fo k = v(e iϕ ) sin kϕdϕ = Fomula (2) can be wien as v(e iϕ ) sin ϕdϕ = λ k () 2k+ d + k v( e iϕ ) sin kϕdϕ, k N, λ() d + v( 3 e iϕ ) sin ϕdϕ. (2) T (, v) = T (, v). (3) Definiion. The ode and lowe ode of a gowh funcion γ ae defined o be he values: ln γ() ln γ() β[γ] = lim, α[γ] = lim ln ln. Definiion 2. The ode and lowe ode of a funcion v Jδ ae defined o be he values β[t (, v)] and α[t (, v)]. 2 The Fouie coefficiens of a funcion v Jδ ae defined by he fomula ([9]) c k (, v) = v(eiθ ) sin kθdθ, k N. Le λ be he full measue of v Jδ, hen ([9]) whee α k = k c k (, v), and + k k 2k C + (, ) c k (, v) = α k k + 2k sin kϕ Im ζ τ k dλ(ζ) + k k c k (, v) = α k k + D + (,) λ k () d, k N, (4) 2k+ sin kϕ τ k Im ζ dλ(ζ) k k C + (,) sin kϕ Im ζ τ k dλ(ζ), (5) whee ζ = τe iϕ. By he definiion of c k (, v) one has c k (, v) 2k v(eiϕ ) sin ϕdϕ, k N. Taking ino accoun (3) we obain T (, v) 2k c k(, v), k N. (6) 3. The main esul of his pape is he following heoem. Theoem. If v SK is a subhamonic funcion on C + of infinie ode wih he full measue λ on he finie sysem of ays L k = {z : ag z = e iθ k, θk = p k q k }; k, N ; p k, q k, N N; p k < q k ; hen is lowe ode equals infiniy. Poof. We assume ha / supp v. By fomulae (5) fo Fouie coefficiens of he funcion v we obain N + c n (, v) = α n n + n n dλ() n+ N n n 2n N n n n dλ()+ n dλ(), n N.
82 K. G. MALYUTIN, T. I. MALYUTINA, I. I. KOZLOVA Assume saisfies C(, ) / supp v. Then we obain c n (, v) = α n n + N n [ ( ) n ( ) n ] dλ(), n N. (7) Applying wice he inegaion by pas in (7), we obain ( c n (, v) = α n n + 2 N Ñ() + n + n N [ ( ) n ( ) Ñ() d + n+ ) n ] Ñ()d, n N, (8) whee Ñ() = λ() 2 d. Denoe by C = N sin θ k. I is clea ha C >. Fom (8) wih n = n l = +2l N q k, l N, we obain c n (, v) n 2C ( Ñ() Ñ() ) + n d α n+ n, n N. (9) If he funcion Ñ() has infinie ode hen he inegal fom he igh hand side of he lae inequaliy is unbounded as because Ñ() d Ñ(), n N, and he ighhand side of his inequaliy can be made abiaily lage by a suiable choice of. By his n+ n n inequaliy and inequaliy (6), fom (9) we obain he equied saemen. If Ñ() has finie ode hen hee exis posiive numbes K > and ρ > such ha Ñ() K ρ fo all >. I is possible o conside non-inege ρ. Then K2 ρ ρ Ñ(2) 2 λ() 2 d λ() 2 d = λ() 2 2, i.e. λ() K2 ρ+ ρ+. In his case one can deduce fom [8] ha hee exiss a funcion g JS of ode ρ and wih full measue λ. Then G = v g Jδ and λ G. Fuhe we need he following lemma. Lemma. If G JS and λ G, hen G(z) = Im f(z), whee f(z) is an enie eal funcion. Poof. Remind [] ha an enie funcion is said o be eal if f(r) R. As he full measue of he funcion G equals zeo hen fom () i follows ha fo any R > G(z) = R G(z,Re iϕ ) G(Re iϕ )dϕ, z C 2 n + (, R). The igh-hand side is a hamonic funcion in he half-disc C + (, R), which is exended by he coninuiy as zeo on he ineval ( R, R). Since R is an abiay posiive numbe, he funcion G(z) is hamonic on he half-plane C +, which is exended by coninuiy as zeo on he eal axis. By he symmey pinciple, his funcion is exended as a hamonic funcion o he boom half-plane. Then hee exiss a hamonic funcion h(z) on he complex plane such ha f(r) = and G(z) = h(z) fo Im z >. Le h (z) be a funcion which is hamoniously conjugaed o he funcion h(z). Then f(z) = h (z) + ih(z) is an enie funcion, eal on he eal axis and h(z) = Im f(z).
ON SUBHARMONIC FUNCTIONS IN THE HALF-PLANE 83 Accoding o Lemma, G(z) = Im f(z), whee f(z) is an enie eal funcion f(z) = n= a nz n. If only a finie numbe a n, hen f(z) is a polynomial, hence G and v have a finie ode, which conadics he assumpion. As c n (, G) = a n n, n N, he inequaliy T (, v) T (, G) T (, g) c n (, G) + O( ρ ) a n n + O( ρ ),, n N, 2n 2 implies ha α[t (, v)] =. REFERENCES. L.A. Rubel, B.A. Taylo, Fouie seies mehod fo meomophic and enie funcions, Bull. Soc. Mah. Fance, 96 (968), 53 96. 2. J.B. Miles, Quoien epesenaions of meomophic funcions, J. d Analyse Mah., 25 (972), 37 388. 3. A.A. Kondayuk, The Fouie seies mehod fo enie and meomophic funcions of compleely egula gowh, Ma. Sb., 6(48) (978), 3(7), 386 48; English ansl. in Mah. USSR-Sb., 35 (979),, 63 84. 4. A.A. Kondayuk, The Fouie seies mehod fo enie and meomophic funcions of compleely egula gowh. II, Ma. Sb., 3(55) (98), (9), 8 32; English ansl. in Mah. USSR-Sb., 4 (982),, 3. 5. A.A. Kondayuk, The Fouie seies mehod fo enie and meomophic funcions of compleely egula gowh. III, Ma. Sb., 2(62) (983), 3, 33 343; English ansl. in Mah. USSR-Sb., 48 (984), 2, 327 338. 6. J.B. Miles, On enie funcions of infinie ode wih adially disibued zeos, Pacif. J. Mah., 8 (979),, 3 57. 7. I.I. Kozlova, Subhamonic funcion of infinie ode in he half-plane, Zbinyk pas In-u maemayky NAN Ukainu, 9 (22), 2, 27 23. (in Russian) 8. A.F.Gishin, Coninuiy and asympoical coninuiy of subhamonic funcions, Mahemaical Physics, Analysis and Geomey, (994), 2, 93 25. (in Russian) 9. K.G. Malyuin, Fouie seies and δ-subhamonic funcions of finie γ-ype in a half-plane, Ma. Sb., 92 (2), 6, 5 7; English ansl. in Sb.: Mah. 92 (2), 6.. B.Ya. Levin, Disibuion of zeos of enie funcions. English evised ediion Ame. Mah. Soc, Povidence, RI, 98. Sae Univesiy of Sumy malyuinkg@yahoo.com Received 5.2.23 Revised 3.4.24