ON THE UNIQUENESS OF MEROMORPHIC FUNCTIONS SHARING THREE WEIGHTED VALUES. Indrajit Lahiri and Gautam Kumar Ghosh

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1 MATEMATIQKI VESNIK 60 (008), 5 3 UDK originalni nauqni rad reearch aer ON THE UNIQUENESS OF MEROMORPHIC FUNCTIONS SHARING THREE WEIGHTED VALUES Indrajit Lahiri and Gautam Kumar Ghoh Abtract. We rove a uniquene theorem for meromorhic function haring three weighted value, a conequence of which a number of reult follow. 1. Introduction, Definition and Reult Let f and g be two non-contant meromorhic function defined in the oen comlex lane C. For a C { } we ay that f, g hare the value a CM (counting multilicitie) if f, g have the ame a-oint with the ame multilicity and we ay that f, g hare the value a IM (ignoring multilicitie) if we do not conider the multilicitie. We denote by E a et of nonnegative real number of finite linear meaure, not necearily the ame at each of it occurrence. By S(r, f) we mean any quantity atifying S(r, f) = o{t (r, f)} a r (r E), where T (r, f) i the Nevanlinna characteritic function of f. By N 0 (r, a; f, g) we denote the reduced counting function of the common a-oint of f and g. If f and g hare 0, 1, IM, we denote by the counting function of thoe zero of f g which are not the zero of f(f 1) and 1/f. For the tandard notation and definition of the value ditribution theory we refer the reader to [1]. Let a 1, a, a 3, a 4 be four element of C. Then the cro-ratio of thee number i defined a (a 1, a, a 3, a 4 ) = (a 1 a 3 )(a a 4 ) (a a 3 )(a 1 a 4 ). If a k = for ome k {1,, 3, 4} then we define the cro-ratio a (a 1 a 3 )(a a 4 ) (a 1, a, a 3, a 4 ) = lim a k (a a 3 )(a 1 a 4 ). If a, b, c, d are ditinct element of C { }, in the aer we denote by L(w) the following bilinear tranformation (w c)(b d) L(w) = (w d)(b c). Clearly L(a) = (a, b, c, d) and a = (a, 1, 0, ). AMS Subject Claification: 30D35 Keyword and hrae: Meromorhic function, weighted haring, uniquene. 5

2 6 I.Lahiri, G. K. Ghoh In 1989 N. Terglane [6] roved the following theorem. Theorem A. Let a, b, c, d be four ditinct element of C { }. Further uoe that f and g are two ditinct noncontant meromorhic function haring b, c, d CM. If N 0 (r, a; f, g) S(r, f) and (a, b, c, d) { 1,, 1 } then f i a bilinear tranformation of g. In 00 H. X. Yi and X. M. Li [7] conidered the roblem of removing the hyothei (a, b, c, d) { 1,, 1 } in Theorem A. They roved the following reult. Theorem B. Let a 1, a, b, c, d be five ditinct element of C { } and f, g be two noncontant meromorhic function haring b, c, d CM. If N 0 (r, a 1 ; f, g) S(r, f) and N 0 (r, a ; f, g) S(r, f) then f g. Theorem C. Let f and g be two ditinct noncontant meromorhic function haring three value b, c, d C { } CM. If N 0 (r, a; f, g) S(r, f) for ome a C { } \ {b, c, d}, then (a, b, c, d) i a rational number and N(r, a; f) = T (r, f)+s(r, f), N(r, a; g) = T (r, g)+s(r, g) and N 0 (r, a; f, g) = 1 T (r, f)+s(r, f), where i a oitive integer. Theorem D. Let f and g be two ditinct noncontant meromorhic function haring three value b, c, d C { } CM. If N 0 (r, a; f, g) = T (r, f) + S(r, f) for ome a C { } \ {b, c, d}, then (a, b, c, d) { 1,, 1 }, f i a bilinear tranformation of g and aume one of the following relation: (I) L(f) = e γ, L(g) = e γ 1, (II) L(f) = 1 + e γ, L(g) = 1 + e γ, (III) L(f) = e γ, L(g) = e γ 1, where γ i a noncontant entire function. Theorem E. Let f and g be two ditinct noncontant meromorhic function haring three value b, c, d C { } CM. If N 0 (r, a; f, g) S(r, f) and N 0 (r, a; f, g) T (r, f) + S(r, f) for ome a C { } \ {b, c, d}, then (a, b, c, d)( 0, 1, 1,, 1 ) i a rational number and N 0 (r, a; f, g) = 1 T (r, f) + S(r, f), where (> 1) i a oitive integer, f i not a bilinear tranformation of g and aume one of the following relation: (I) L(f) = eγ 1 e (+1)γ 1, L(g) = e γ 1 e (+1)γ 1 + 1, (II) L(f) = e(+1)γ 1 e (+1 )γ 1, L(g) = e (+1)γ 1 e (+1 )γ 1 e γ 1 (III) L(f) = e (+1 )γ 1, L(g) = e γ 1 e (+1 )γ 1 where γ i a noncontant entire function and , 1,

3 On the uniquene of meromorhic function... 7 Conidering the following examle we ee that in Theorem D the trength of haring of none of the value can be weakened from CM to IM. Examle 1.1. Let f = 4(1 ez ) 1 3e z and g = (1 ez ) 3. Then f and g hare 0 1 3ez IM and 1, CM. Since f = (1 + ez ) 1 3e z and g = (1 + ez )(4e z e z 1) 1 3e z, we ee that N 0 (r, ; f, g) = T (r, f) + S(r, f) but the concluion of Theorem D doe not hold. Examle 1.. Let f = ez 3 1 3e z and g = ez (e z 3) 1 3e z. Then f, g hare 0, CM and 1 IM. Since f + 1 = (1 + ez ) 1 3e z and g + 1 = (1 + ez )(1 + e z 4e z ) 1 3e z, we ee that N 0 (r, 1; f, g) = T (r, f) + S(r, f) but the concluion of Theorem D doe not hold. Examle 1.3. Let f = 1 3ez 1 3ez 4(1 e z and g = ) (1 e z. Then f, g hare 0, 1 ) 3 CM and IM. Since f 1 = 1 + ez 4(1 e z ) and g 1 = (1 + ez )(1 4e z + e z ) (1 e z ) 3, we ee that N 0 (r, 1 ; f, g) = T (r, f) + S(r, f) but the concluion of Theorem D doe not hold. Interchanging f and g in the above examle we ee that in Theorem E alo CM haring of value cannot be relaced by IM haring of value. In the aer we invetigate the oibility of relaxing the nature of haring the three value. To thi end we ue the idea of weighted value haring which meaure how cloe a value i being hared CM or being hared IM. We alo intend to rove a ingle uniquene theorem which include all the reult mentioned above. In the following definition we exlain the idea of weighted value haring. Definition 1.1. [, 3] Let k be a nonnegative integer or infinity. For a C { }, we denote by E k (a; f) the et of all a-oint of f where an a-oint of multilicity m i counted m time if m k and k + 1 time if m > k. If E k (a; f) = E k (a; g), we ay that f and g hare the value a with weight k. The definition imlie that if f and g hare a value a with weight k then z 0 i a zero of f a with multilicity m( k) if and only if it i a zero of g a with multilicity m( k) and z 0 i a zero of f a with multilicity m(> k) if and only if it i a zero of g a with multilicity n(> k) where m i not necearily equal to n. We write f, g hare (a, k) to mean that f, g hare the value a with weight k. Clearly if f, g hare (a, k) then f, g hare (a, ) for all integer, 0 < k. Alo we note that f, g hare a value a IM or CM if and only if f, g hare (a, 0) or (a, ) reectively. We now tate the main reult of the aer. Theorem 1.1. Let f and g be two ditinct noncontant meromorhic function haring (0, k 1 ), (1, k ), (, k 3 ), where k 1 k k 3 > k 1 + k + k 3 +. If N 0 (r, a; f, g)

4 8 I.Lahiri, G. K. Ghoh S(r, f) for ome a {0, 1, } then one of the following hold: (I) f = eγ 1 e (+1)γ 1, g = e γ 1 e (+1)γ 1 and a = + 1, (II) f = e(+1)γ 1 e (+1 )γ 1, g = e (+1)γ 1 e (+1 )γ 1 and a = , e γ 1 (III) f = e (+1 )γ 1, g = e γ 1 e (+1 )γ 1 and a = 1, where and are oitive integer with 1 and, + 1 are relatively rime and γ i a noncontant entire function. Further N 0 (r, a; f, g) = 1 T (r, f) + S(r, f). Theorem 1.. Let f and g be two ditinct noncontant meromorhic function haring (b, k 1 ), (c, k ), (d, k 3 ), where k 1 k k 3 > k 1 + k + k 3 + and b, c, d are three ditinct element of C { }. If N 0 (r, a; f, g) S(r, f) for ome a C { } \ {b, c, d} then one of the following hold: (I) L(f) = eγ 1 e (+1)γ 1, L(g) = e γ 1 e (+1)γ 1 (II) L(f) = e(+1)γ 1 e (+1 )γ 1, L(g) = e (+1)γ 1 e (+1 )γ 1 e γ 1 (III) L(f) = e (+1 )γ 1, L(g) = e γ 1 e (+1 )γ 1 + 1, , 1, where and are oitive integer with 1 and, + 1 are relatively rime and γ i a noncontant entire function. Further N 0 (r, a; f, g) = 1 T (r, f) + S(r, f). Following corollarie of Theorem 1. imrove Theorem A Theorem E reectively. Corollary 1.1. Theorem A hold even if f, g hare (b, k 1 ), (c, k ), (d, k 3 ), where k 1 k k 3 > k 1 + k + k 3 +. Corollary 1.. Theorem B hold even if f, g hare (b, k 1 ), (c, k ), (d, k 3 ), where k 1 k k 3 > k 1 + k + k 3 +. Corollary 1.3. Theorem C hold even if f, g hare (b, k 1 ), (c, k ), (d, k 3 ), where k 1 k k 3 > k 1 + k + k 3 +. Corollary 1.4. Theorem D hold even if f, g hare (b, k 1 ), (c, k ), (d, k 3 ), where k 1 k k 3 > k 1 + k + k 3 +.

5 On the uniquene of meromorhic function... 9 Corollary 1.5. Theorem E hold even if f, g hare (b, k 1 ), (c, k ), (d, k 3 ), where k 1 k k 3 > k 1 + k + k Lemma In thi ection we tate three lemma, which are neceary to rove the main reult. Lemma.1. [4] Let f and g be two ditinct non-contant meromorhic function haring (0, k 1 ), (1, k ), and (, k 3 ), where k j (j = 1,, 3) are oitive integer atifying k 1 k k 3 > k 1 + k + k 3 +. If lim u r,r E T (r, f) > 1 then one of the following relation hold: (i) f + g 1, (ii) (f 1)(g 1) 1, and (iii) fg 1. Lemma.. [4] Let f and g be two ditinct non-contant meromorhic function haring (0, k 1 ), (1, k ) and (, k 3 ), where k j (j = 1,, 3) are oitive integer atifying k 1 k k 3 > k 1 + k + k 3 +. If 0 < lim u r,r E T (r, f) 1 then one of the following hold: (i) f = eγ 1 e (+1)γ 1, g = e γ 1 (1 ), e (+1)γ 1 (ii) f = e(+1)γ 1 e (+1 )γ 1, g = e (+1)γ 1 (1 ), e (+1 )γ 1 e γ 1 (iii) f = e (+1 )γ 1, g = e γ 1 (1 ), e (+1 )γ 1 where and ( ) are oitive integer uch that, + 1 are relatively rime and γ i a noncontant entire function. Lemma.3. [5] Let f be a non-contant meromorhic function and m i=0 R(f) = a if i n j=0 b jf j be a non-contant irreducible rational function in f with contant coefficient {a i } and {b j } atifying a m 0 and b n 0. Then T (r, R(f)) = max{m, n}t (r, f) + O(1). 3. Proof of theorem and corollarie Proof of Theorem 1.1. Since N 0 (r, a; f, g) and N 0 (r, a; f, g) S(r, f), we ee that lim u r,r E T (r, f) > 0. We now conider the following two cae.

6 30 I.Lahiri, G. K. Ghoh Cae I. Let lim u r,r E T (r, f) > 1. Then the three oibilitie of Lemma.1 come u for conideration. Let f + g 1. Since N 0 (r, a; f, g) S(r, f), we ee that a = 1. Alo 0 and 1 are Picard excetional value of f and g. So there exit a noncontant entire 1 function γ uch that f = e γ + 1 and g = 1 e γ, which i oibility (I) of the + 1 theorem for = 1. Let (f 1)(g 1) 1. Since N 0 (r, a; f, g) S(r, f), we ee that a =. Alo 1 and are Picard excetional value of f and g. So there exit a noncontant entire function γ uch that f = e γ + 1 and g = e γ + 1, which i oibility (II) of the theorem for = 1. Let fg 1. Since N 0 (r, a; f, g) S(r, f), we ee that a = 1. Alo 0 and are Picard excetional value of f and g. So there exit a noncontant entire function γ uch that f = e γ and g = e γ, which i oibility (III) of the theorem for = 1. Alo we ee in view of Lemma.3 that N 0 (r, a; f, g) = N(r, 1; e γ ) = T (r, e γ ) + S(r, e γ ) = T (r, f) + S(r, f). T (r, f) 1. Then we conider the three oibili- Cae II. Let 0 < lim u tie of Lemma.. and Let r,r E f = eγ 1 e (+1)γ 1 = 1 + eγ + e γ + + e ( 1)γ 1 + e γ + e γ + + e γ g = e γ 1 e (+1)γ 1 = 1 + e γ + e γ + + e ( 1)γ 1 + e γ + e γ + + e γ. Since N 0 (r, a; f, g) S(r, f), we ee from above that a = +1, which i attained at the root of e γ 1 = 0. Thi i oibility (I) of the theorem. and Let f = e(+1)γ 1 e (+1 )γ 1 = 1 + eγ + e γ + + e γ 1 + e γ + e γ + + e ( )γ g = e (+1)γ 1 e (+1 )γ 1 = 1 + e γ + e γ + + e γ 1 + e γ + e γ + + e. ( )γ Since N 0 (r, a; f, g) S(r, f), we ee from above that a = +1 +1, which i attained at the root of e γ 1 = 0. Thi i oibility (II) of the theorem. Let e γ 1 f = e (+1 )γ 1 = 1 + e γ + e γ + + e ( 1)γ eγ 1 + e γ + e γ + + e ( )γ

7 and On the uniquene of meromorhic function g = e γ 1 e (+1 )γ 1 = eγ 1 + e γ + e γ + + e ( 1)γ 1 + e γ + e γ + + e ( )γ. Since N 0 (r, a; f, g) S(r, f), we ee from above that a = 1, which i attained at the root of e γ 1 = 0. Thi i oibility (III) of the theorem. Again from above we ee in view of Lemma.3 N 0 (r, a; f, g) = N(r, 1; e γ ) = T (r, e γ ) + S(r, e γ ) = 1 T (r, f) + S(r, f). Thi rove the theorem. Proof of Theorem 1.. From the hyothee of the theorem we ee that L(f) and L(g) hare (0, k ), (1, k 1 ), (, k 3 ). Alo in view of Lemma.3 we get N 0 (r, L(a); L(f), L(g)) = N 0 (r, a; f, g) S(r, f) = S(r, L(f)). Since k 1 and k are interchangeable, Theorem 1. follow from Theorem 1.1 alied to L(f) and L(g). Thi rove the theorem. Proof of Corollary 1.1. From the hyothee we ee that the oibilitie of Theorem 1. occur only for = 1. Hence L(f) i a bilinear tranformation of L(g). Therefore f i a bilinear tranformation of g. Proof of Corollary 1.. If f g, by Theorem 1. we ee that L(a 1 ) = L(a ) and o a 1 = a, which contradict the hyothee. Hence f g. Proof of Corollary 1.3. By the hyothee and Theorem 1. we ee that L(a) = (a, b, c, d) i a rational number. and Alo in view of Theorem 1. and Lemma.3 we can verify that N(r, a; f) = N(r, L(a); L(f)) = T (r, L(f)) + S(r, L(f)) = T (r, f) + S(r, f) N(r, a; g) = N(r, L(a); L(g)) = T (r, L(g)) + S(r, L(g)) = T (r, g) + S(r, g). Further by Theorem 1. we obtain N 0 (r, a; f, g) = 1 T (r, f) + S(r, f). Proof of Corollary 1.4. By the hyothee we ee that = 1 in Theorem 1. and o the corollary follow. Proof of Corollary 1.5. By the hyothee we get > 1 in Theorem 1.. Hence the corollary follow.

8 3 I.Lahiri, G. K. Ghoh REFERENCES [1] W. K. Hayman, Meromorhic Function, The Clarendon Pre, Oxford [] I. Lahiri, Weighted haring and uniquene of meromorhic function, Nagoya Math. J., 161 (001), [3] I. Lahiri, Characteritic function of meromorhic function haring three value with finite weight, Comlex Variable Theory Al., 50, 1 (005), [4] I. Lahiri and P. Sahoo, Uniquene of meromorhic function haring three weighted value, Bull. Malayian Math. Sci. Soc., () 31 (1) (008), [5] A. Z. Mohonko, On the Nevanlinna characteritic of ome meromorhic function, Funct. Anal. Al., 14 (1971), [6] N. Terglane, Identitätätze in C Meromorher Funktionen al Ergebni von Werteteilung, Dilomarbeit, RWTH Aachen (1989). [7] H. X. Yi and X. M. Li, On a reult of Terglane concerning uniquene of meromorhic function, Indian J. Pure Al. Math., 33, 1 (00), (received , in revied form ) Deartment of Mathematic, Univerity of Kalyani, Wet Bengal 74135, India. indr9431@dataone.in Deartment of Mathematic, G.M.S.M. Mahavidyalaya, Birewarur, South 4 Pargana, India. g80g@rediffmail.com

UNIQUENESS OF MEROMORPHIC FUNCTIONS SHARING THREE VALUES

Kragujevac Journal of Mathematics Volume 38(1) (2014), Pages 105 123. UNIQUENESS OF MEROMORPHIC FUNCTIONS SHARING THREE VALUES ARINDAM SARKAR 1 AND PAULOMI CHATTOPADHYAY 2 Abstract. We prove four uniqueness