AVERAGING OF ENTIRE FUNCTIONS OF IMPROVED REGULAR GROWTH WITH ZEROS ON A FINITE SYSTEM OF RAYS R.V. Khats

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1 ÂIÑÍÈÊ ÍÀÖIÎÍÀËÜÍÎÃÎ ÓÍIÂÅÐÑÈÒÅÒÓ ËÜÂIÂÑÜÊÀ ÏÎËIÒÅÕÍIÊÀ Ôiçèêî-ìàòåìàòè íi íàóêè Âèï , (211) c. 114 JOURNAL OF NATIONAL UNIVERSITY LVIVSKA POLITECHNIKA Physical & mahemaical sciences Vol.718 No 718, (211) 1 14 AVERAGING OF ENTIRE FUNCTIONS OF IMPROVED REGULAR GROWTH WITH ZEROS ON A FINITE SYSTEM OF RAYS R.V. Khas Ivan Fanko Dohobych Sae Pedagogical Univesiy, Insiue of Physics, Mahemaics Infomaics 3 Syiska S., 821, Dohobych, Ukaine (Received 7 21.) Using a Fouie seies mehod fo enie funcions, we find an asympoics of aveaging of enie funcions of impoved egula gowh wih zeos on a finie sysem of ays. Key wods: enie funcion of impoved egula gowh, Fouie coefficiens, finie sysem of ays. 2 MSC: 3D15 UDK: Inoducion main esul In [1, 2] (see also [3]) he class of enie funcions of impoved egula gowh was inoduced a cieia fo his egulaiy in he sense of zeo disibuion ae esablished, when he zeos ae locaed on a finie sysem of ays. In connecion wih he sudy of enie funcions of impoved egula gowh wih zeos on abiay sysem of ays, in [4] was poved he following saemen. Theoem A. Le f be an enie funcion of ode (, + ) wih he indicao h le fo some 1 (, ) hee exiss an excepional se U C such ha log f(z) z h(φ)+o( z 1 ), U z e iφ, (1) U can be coveed by a sysem of paiwise disjoin disks U k {z : z a k < τ k }, k N, saisfying τ k < +, τ k log τ k < +. k N k N Then hee exiss 2 (, ) such ha J f (φ) : 1 log f(e iφ ) d h(φ) + o(2 ) (2) as + unifomly wih espec o φ [, ]. In he pesen pape, using he Fouie seies mehod [5] fo he logaihm of he modulus of an enie funcion we shall pove he analog of Theoem A fo enie funcions of impoved egula gowh wih zeos on a finie sysem of ays. We emind he eade ha an enie funcion f is said o be of impoved egula gowh (see [1, 2]), if fo some (, + ), 1 (, ), some -peiodic -igonomeically convex funcion h(φ) hee exiss an excepional se U C such ha elaion (1) holds U can be coveed by a sysem of disks wih finie sum of adii. Remak ha if f is an enie funcion of impoved egula gowh, hen i has [1] he ode indicao funcion h. Le us fomulae ou main esul. Theoem 1. If an enie funcion f of ode (, + ) wih zeos on a finie sysem of ays {z : ag z }, j {1,..., m}, ψ 1 < ψ 2 <... < ψ m <, is of impoved egula gowh, hen fo some 2 (, ) he elaion (2) holds unifomly wih espec o φ [, ]. 2. Peliminaies Le f be an enie funcion wih f() 1, (λ n ) n N be a sequence of is zeos, p is he smalles inege fo which he seies n1 λ n p 1 conveges. By c k (, log f ) 1 c k (, J f ) 1 e ikφ log f(e iφ ) dφ, k Z, e ikφ J f (φ) dφ, k Z, >, we denoe he Fouie coefficiens of he funcions log f(e iφ ) Jf (φ), especively. Lemma 1. If an enie funcion f of ode (, + ) wih zeos on a finie sysem of ays {z : ag z }, j {1,..., m}, ψ 1 < ψ 2 <... < ψ m <, is of impoved egula gowh, hen hee exiss 3 (, ) such ha he asympoic elaion c k (, log f ) α k ) + o(3 k 2, +, (3) holds unifomly fo k Z, whee 2 k 2 α k : 1 j e ikψj, e ikφ h(φ) dφ j [, + ), Ìàòåìàòèêà c R.V. Khas, 211

2 Aveaging of enie funcions of impoved egula gowh wih zeos on a finie sysem of ays fo a noninege, 2 k 2 j e ikψj, k p, τ f e iθ f α k j e iψj, k p, 2 4, k p, Q 2, k p, if is an inege. Poof. Le is noninege. Then [1] an enie funcion f, f() 1, can be epesened in he fom f(z) e Q(z) m L j (z), (4) whee Q(z) : ν k1 Q kz k is a polynomial of degee ν < L j (z) is he Weiesass canonical poduc of genus p, p [] < < p, consuced in zeos of he funcion f which lie on he ay {z : ag z }. Since f is an enie funcion of impoved egula gowh, hen [2] fo some 4 (, ) evey j {1,..., m} n j () j + o( 4 ), +, j [, + ), (5) whee n j () is he numbe of zeos of he funcion f fom he disk {z : z }, which ae concenaed on a ay {z : ag z }. Moeove, he indicao h of an enie funcion f of impoved egula gowh of noninege ode has he fom ([1]) h(φ) h j (φ), whee h j (φ) is a -peiodic funcion such ha on [, + ) Theefoe, α k 1 h j (φ) π j sin π cos (φ π). π j sin π 2 k 2 + e ikφ cos (φ π) dφ j e ikψj, k Z. Fuhe, in view of (4), we have (see [3,6]) c k (, log f ) c k (, log f ), k 1, (6) c (, log f ) N j (), N j () n j () d, (7) c k (, log f ) 1 2 Q k k + < λ n, ag λ n c k (, log f ) + < λ n, ag λ n [ ( ) k 1 k p, ( ) ] k λn λ n >, ag λ n e ikψj ( ) k +, (8) ( ) k λn, k p. (9) e ikψj Using (5), fom (8), inegaing by pas, fo 1 k p we obain c k (, log f ) 1 2 Q k k + ( ( ) ( ) ) k k dn j () e ikψj k k 2 k Q k k + n j () k+1 d + k k k 1 n j () d e ikψj j e ikψj + o(4 ) k 2 α k ) + o(4 k 2, +. (1) Similaly, using fomulas (5) (9), fo k p we ge c k (, log f ) + ( ) k ( ) k dnj () + dn j () e ikψj 1 k k + 2 k 2 n j () k+1 d k k j e ikψj + o(4 ) k 2 k 1 n j () d e ikψj α k ) + o(4 k 2, +. (11) Fom (1), (11), (6) (7), i follows (3). Le now N. Then [2] an enie funcion f is of fom (4), whee Q(z) is a polynomial of degee ν, p is he smalles inege such ha 1 λ n p 1 < +, L j (z) is a Weiesass canonical poduc of genus p, p o p, consuced by he zeos of f which lie on a ay {z : ag z }. Since f is an enie funcion of impoved egula gowh, hen [2] fo some Mahemaics 11

3 R.V. Khas 4 (, ) evey j {1,..., m} he elaion (5) holds,, in addiion, fo some δ f C 5 (, ) < λ n λ n δ f + o( 5 ), +. (12) Besides, he indicao h of an enie funcion f of impoved egula gowh of ode N is defined by he fomula ([2]) τ f cos(φ + θ f ) + h j (φ), p, h(φ) Q cos φ, p, (13) whee τ f δ f / + Q, θ f ag(δ f / + Q ) h j (φ) is a -peiodic funcion such ha on [, + ) h j (φ) j (π φ+ ) sin (φ ) j cos (φ ). Fis, le p. Then, accoding o (13), we ge α k 1 j + j 2 k 2 e ikφ τ f cos(φ + θ f ) dφ+ e ikφ (π φ + ) sin (φ ) dφ + e ikφ cos (φ ) dφ j e ikψj, k, α τ f e iθ f 2 4 j e iψj. Fuhe, (see [2, 3, 6]) he fomula (8) holds fo 1 k < p, fomula (9) is ue fo k p,, in paicula, fo k p we have c (, log f ) 1 2 Q 2 whee I(j) < λ n, ag λ n < λ n ( λ n ) 2 ( ) λn e iψj. I(j), (14) Thus, in he same way as in he case of noninege, fo 1 k < p k p he asympoic elaion (3) holds. Le s conside he case k p. Taking ino accoun (5), we obain I(j) e iψj dn j () j 2 e iψj + o( 4 ) (15) as +. Combining (14), (15) (12), we ge c (, log f ) 2 (Q + δ f /) + o( 5 ) τ f e iθ f 2 4 j e iψj + o( 4 ) 4 j e iψj + o( 6 ) α + o( 6 ), +, < 6 <. Hence, in he case k p we also obain (3). Fo ohe values of k he equied saemen follows fom elaions (6) (7). Now conside he case p+1. Taking ino accoun (13), we ge α k 1 α 1 e ikφ Q cos φ dφ, k, e iφ Q cos φ dφ Q 2. Besides, (see [3, 6]) he fomula (8) is fulfilled fo 1 k p, fomula (9) is ue fo k > p, in paicula, fo k p we have c (, log f ) 1 2 Q 2 2 λ n >, ag λ n ( I(j) ) e iψj, whee I(j) is defined above. Since [2] in his case he elaion (5) holds wih j, hen likewise as in he case p, fo k Z, k p, he elaion (3) is valid. Le now k p. In view of (15), I(j) o( 4 ) as +, i is easy o show ha λ n >, ag λ n ( as +. Theefoe ) + e iψj e iψj c (, log f ) 1 2 Q +o( 4 ) α +o( 4 ), The poof of Lemma 1 is hus compleed. dn j () o( 4 ) +. Coollay 1. Unde he condiions of Lemma 1, we have c k (, J f ) α k + o(3 ) k 2, +, 12 Ìàòåìàòèêà

4 Aveaging of enie funcions of impoved egula gowh wih zeos on a nie sysem of ays fo some 3 (, ) unifomly in k Z. Indeed, ([5, p. 112]) c k (, J f ) c k (, log f ) whence he equied poposiion follows. 3. Poof of Theoem 1 d, k Z, Indeed, a Fouie seies of he funcion Jf (φ) h(φ) is he seies ) (c k (, J f ) α k e ikφ. k Z Then, accoding o Coollay 1, we ge k Z ) (c k (, J f ) α k e ikφ o( 3 ), +, fo some 3 (, ) unifomly wih espec o φ [, ], he Theoem 1 is poved. Coollay 2. Le he hypoheses of Theoem 1 be saised. Then fo some 2 (, ) 1 J f (φ) d 2 h(φ) + o(2 ), unifomly wih espec o φ [, ]. +, Refeences [1] Âèííèöüêèé Á.Â., Õàöü Ð.Â. Ïðî ðåãóëÿðíiñòü çðîñòàííÿ öiëî ôóíêöi íåöiëîãî ïîðÿäêó ç íóëÿìè íà ñêií åííié ñèñòåìi ïðîìåíiâ // Ìàòåì. ñòóäi , 1. Ñ [2] Khas' R.V. On enie funcions of impoved egula gowh of inege ode wih zeos on a nie sysem of ays // Ìàòåì. ñòóäi , 1. Ñ [3] Õàöü Ð.Â. Öiëi ôóíêöi ïîêðàùåíîãî ðåãóëÿðíîãî çðîñòàííÿ: Äèñ.... êàíä. ôiç.-ìàò. íàóê. Äðîãîáè ñ. [4] Vynnys'kyi B.V., Khas' R.V. On asympoic popeies of enie funcions, simila o he enie funcions of compleely egula gowh // Âiñíèê ÍÓ "Ëüâiâñüêà ïîëiòåõíiêà". Ñåðiÿ ôiç.-ìàò. íàóêè.?.?,?. Ñ.??. [5] Êîíäðàòþê À.À. Ðÿäû Ôóðüå è ìåðîìîðôíûå ôóíêöèè. Ëüâîâ: Âûùà øêîëà, ñ. [6] Õàöü Ð.Â. Ïðî êîåôiöi¹íòè Ôóð'¹ îäíîãî êëàñó öiëèõ ôóíêöié // Ìàòåì. ñòóäi , 1. Ñ

5 R.V. Khas' ÓÑÐÅÄÍÅÍÈÅ ÖÅËÛÕ ÔÓÍÊÖÈÉ ÓËÓ ØÅÍÍÎÃÎ ÐÅÃÓËßÐÍÎÃÎ ÐÎÑÒÀ Ñ ÍÓËßÌÈ ÍÀ ÊÎÍÅ ÍÎÉ ÑÈÑÒÅÌÅ ËÓ ÅÉ Ð.Â. Õàöü Äðîãîáèöêèé ãîñóäàðñòâåííûé ïåäàãîãè åñêèé óíèâåðñèòåò èìåíè Èâàíà Ôðàíêî, Èíñòèòóò ôèçèêè, ìàòåìàòèêè è èíôîðìàòèêè óë. Ñòðèéñêàÿ, 3, Äðîãîáû, 821, Óêðàèíà Ñ ïîìîùüþ ìåòîäà ðÿäîâ Ôóðüå äëÿ öåëûõ ôóíêöèé íàéäåíà àñèìïòîòèêà óñðåäíåíèé öåëûõ ôóíêöèé óëó øåííîãî ðåãóëÿðíîãî ðîñòà ñ íóëÿìè íà êîíå íîé ñèñòåìå ëó åé. Êëþ åâûå ñëîâà: öåëàÿ ôóíêöèÿ óëó øåíîãî ðåãóëÿðíîãî âîçðîñòàíèÿ, êîýôôèöèåíòû Ôóðüå, êîíå íàÿ ñèñòåìà ëó åé. 2 MSC: 3D15 ÓÄÊ: ÓÑÅÐÅÄÍÅÍÍß ÖIËÈÕ ÔÓÍÊÖIÉ ÏÎÊÐÀÙÅÍÎÃÎ ÐÅÃÓËßÐÍÎÃÎ ÇÐÎÑÒÀÍÍß Ç ÍÓËßÌÈ ÍÀ ÑÊIÍ ÅÍÍIÉ ÑÈÑÒÅÌI ÏÐÎÌÅÍI Ð.Â. Õàöü Äðîãîáèöüêèé äåðæàâíèé ïåäàãîãi íèé óíiâåðñèòåò iìåíi Iâàíà Ôðàíêà, Iíñòèòóò ôiçèêè, ìàòåìàòèêè òà iíôîðìàòèêè âóë. Ñòðèéñüêà, 3, 821, Äðîãîáè, Óêðà íà Çà äîïîìîãîþ ìåòîäó ðÿäiâ Ôóð'¹ äëÿ öiëèõ ôóíêöié, çíàéäåíî àñèìïòîòèêó óñåðåäíåíü öiëèõ ôóíêöié ïîêðàùåíîãî ðåãóëÿðíîãî çðîñòàííÿ ç íóëÿìè íà ñêií åííié ñèñòåìi ïðîìåíiâ. Êëþ îâi ñëîâà: öiëà ôóíêöiÿ ïîêðàùåíîãî ðåãóëÿðíîãî çðîñòàííÿ, êîåôiöi¹íòè Ôóð'¹, ñêií- åííà ñèñòåìà ïðîìåíiâ. 2 MSC: 3D15 UDK: