Математичнi Студiї. Т.48, №2 Matematychni Studii. V.48, No.2 УДК 517.547 A. I. Bandura, M. M. Sheremeta BOUNDED l-INDEX AND l-M-INDEX AND COMPOSITIONS OF ANALYTIC FUNCTIONS A. I. Bandura, M. M. Sheremeta. Bounded l-index and l-M-index and compositions of analytic functions, Mat. Stud. 48 (2017), 180–188. We partially proved a conjecture from Mat. Stud. 47 (2017), no.2, 207{210: for an entire function f the function H(z) = f(1=(1−z)n), n 2 N, is of bounded l-index in C n f0g with l(jzj) = β=(1 − jzj)n+1, β > 1, if and only if f is of bounded index. Also the boundedness of l-M- index of the function H is investigated. For arbitrary entire functions f and g the boundedness of the l-M-index of the function F (z) = f(g(z)) is studied with respect to boundedness of the 0 fj j j j g M-index of a function f with l(r) = Mg(r);Mg(r) = max g(z) : z = r : 1. Introduction. In the paper, we consider compositions of such types H(z)=f((1−z)−n); n 2 N; and F (z) = f(g(z)); where f and g are entire functions. There are presented investi- gations of boundedness of l-index and l-M-index for these composite functions, where l is a positive continuous function in some domain G ⊆ C: We need some definitions and notations. Let G be an arbitrary domain in C and l be a positive and continuous function in G such that for all z 2 G β l(z) > ; β = const > 1: (1) distfz; @Gg An analytic function f in G is said ([17, 21]) to be of bounded l-index if there exists N 2 Z+ such that for all n 2 Z+ and z 2 G { } jf (n)(z)j jf (k)(z)j ≤ max : 0 ≤ k ≤ N : (2) n!ln(z) k!lk(z) The least such integer is called the l-index of f and is denoted by N(f; l; G). Let 0 < R ≤ +1, DR = fz : jzj < Rg and l be a positive continuous function on [0;R), which satisfies β l(r) > ; β = const > 1: (3) R − r If G = DR then an analytic in DR function f [21] is of bounded l-index if there exists N 2 Z+ such that for all n 2 Z+ and z 2 DR inequality (2) holds with l(jzj) instead of l(z): If R = +1 (i. e. f is an entire function) then condition (3) is unnecessary. We remark also that if f is an entire function and l(jzj) ≡ 1 then f is said [18] to be of bounded index. If R = 1 then D = D1: 2010 Mathematics Subject Classification: 30D05, 30D15. Keywords: analytic function; enitre function; bounded l-index; bounded index; bounded M-index; bounded l-M-index; composition of functions; growth estimate; punctured plane; disc; maximum modulus. doi:10.15330/ms.48.2.180-188 ⃝c A. I. Bandura, M. M. Sheremeta, 2017 BOUNDED l-INDEX AND l-M-INDEX AND COMPOSITIONS OFANALYTIC FUNCTIONS181 l(z) r l(z) For r 2 [0; β] we put λ1(r) = inff : jz−z0j ≤ ; z0 2 Gg and λ2(r) = supf : jz− l(z0) l(z0) l(z0) r z0j ≤ ; z0 2 Gg: By Qβ(G) we denote the class of positive coninuous functions l satisfying l(z0) (1) and for some r0 2 [0; β] 0 < λ1(r0) ≤ λ2(r0) < +1: (4) We remark that if (4) holds for some r0 2 [0; β] then the inequality is valid for all r 2 [0; β]: Recently, there was suggested the following conjecture. Conjecture 1 ([1]). For an entire function f the function F (z) = f ((1 − z)−n), n 2 N, is of bounded l-index in D with l(jzj) = β(1 − jzj)−n−1, β > 1, if and only if f is of bounded index. Now we consider a more general hypothesis: Conjecture 2 (M. M. Sheremeta). Let f be an entire function and q q q g(z) = 1 + 2 + ::: + p + q ; p 2 N; q 2 C; q =6 0: (1 − z)p (1 − z)p−1 1 − z p+1 j 1 A composite function F = f◦g in D is of bounded l-index in Cnf0g with l(jzj) = β(1−jzj)−p−1 if and only if f is of bounded index. In the paper, Conjecture 2 is partially proved (Theorems 5, 6, 7). The similar problem we also consider for bounded l-M-index (Theorem 9 and 10). Note that there are few papers about boundedness of l-index for compositions of entire functions of one variable ( [20,21]). The growth of a composition of entire functions of finite order is examined in [22]. The L-index in direction of some compositions of entire functions of several variables and its properties are studied in [4–6,13]. But the l-M-index of composite functions are not investigated yet. 2. Auxiliary propositions. We need the following assertions. Theorem 1 (Theorem 1.5, [21]). Let β > 1 and l 2 Qβ(G): An analytic function f in the domain G is of bounded l-index if and only if there exist numbers m 2 Z+ and C > 0 such that for each z 2 G { } jf (m+1)(z)j jf (k)(z)j ≤ C max : 0 ≤ k ≤ m : (5) lm+1(z) lk(z) Theorem 1 is an analog of known Hayman’s Theorem [16] for analytic functions. Theorem 2 (Theorem 2.2, [21]). Let G be an arbitrary domain in C and a domain D ⊂ G be such that distf@D; @Gg > 0: Let β > 1 and l be a positive continuous functions in G such that l(z) ≥ β=d for all z 2 G: If a function f is analytic in G then f is of bounded l-index in D: 1 ≤ 1 2 ≤ Theorem 3 (Theorem 1.6, [21]). Let β > 1; β < θ1 θ2 < + ; l Qβ(G) and θ1l(z) l∗(z) ≤ θ2l(z) for all z 2 G: An analytic in the domain G function f is of bounded l∗-index if and only if it is of bounded l-index. Remark 1. Actually, in the proof of Theorem 3 there was proved that if l1; l2 are positive continuous functions in G, l1(z) ≤ l2(z) for all z 2 G; f is analytic function in G then N(f; l2;G) ≤ N(f; l1;G): 182 A. I. BANDURA, M. M. SHEREMETA ≤ 1 2 D D Theorem 4 (Theorem 3.3, [21]). Let 0 < R + ; lR Qβ( R) and an analytic in R r ! function f is of bounded l-index. Then ln M(r; f) = O( 0 l(t)dt); r R; where M(r; f) = maxfjf(z)j: jzj = rg: Some analogs of Theorems 1–4 are also obtained for analytic functions in a polydisc [11, 12], in the unit ball [7, 8]. The similar results are known for entire functions in Cn ( [3,9,10]). 3. Bounded l-index of some compositions. Taking into account Conjecture 2 it is possible to prove such a theorem. q1 q2 Theorem 5. Let f be an entire function. A function f( zp + zp−1 + ::: + qp+1) has bounded l-index in C n f0g with l(jzj) = βjzj−p−1 if and only if the function f has bounded index, where p 2 N; qj 2 C; q1 =6 0: Proof. If p = 1 and q2 = 0 then the same proposition is obtained in [21, p. 99]. We will deduce the statement for arbitrary p 2 N: q1 q2 Let F (z) = f( zp + zp−1 + ::: + qp+1): It is easy to check that ( ) ( ) q q q p q (p − 1) q F 0(z) = −f 0 1 + 2 + ::: + q 1 + 2 + ::: + p ; zp zp−1 p+1 zp+1 zp z2 ( ) ( ) q q q p q (p − 1) q k F (k)(z) = f (k) 1 + 2 + ::: + q 1 + 2 + ::: + p + zp zp−1 p+1 zp+1 zp z2 k−1 ( )( ) X q q c c + f (j) 1 + 2 + ::: + q j;pj+k + ::: + j;j+k (6) zp zp−1 p+1 zpj+k zj+k j=1 and ( ) 0 p+1 0 q1 q2 − F (z)z f p + p−1 + ::: + qp+1 = p−1 ; z z q1p + q2(p − 1)z + ::: + qpz ( ) ( ) q q zp+1 k (k) 1 2 − k (k) f p + p−1 + ::: + qp+1 = ( 1) F (z) p−1 + z z q1p + q2(p − 1)z + ::: + qpz − Xk 1 zpk+jQ (z) + F (j)(z) j;k ; (7) (q p + q (p − 1)z + ::: + q zp−1)2k−1 j=1 1 2 p where Qj;k(z) are some polynomials with degrees depending from j and k: Equalities (6)–(7) can be proved by the method of mathematical induction (see a similar proof in [13]). Let f be an entire function of bounded index. Then from (6) by Theorem 1 we have jF (m+1)(z)jjzj(p+1)(m+1) ≤ ( ) q q ≤ jf (m+1) 1 + 2 + ::: + q j q p + q (p − 1)z + ::: + q zp−1 m+1 + zp zp−1 p+1 1 2 p m ( ) X q q ( ) + f (j) 1 + 2 + ::: + q jc jjzjp(m+1−j) + ::: + jc jjzj(m+1)p−j ≤ zp zp−1 p+1 j;pj+m+1 j;j+m+1 j=1 ! Xm ( ) p(m+1−j) (m+1)p−j ≤ C + jcj;pj+m+1jjzj + ::: + jcj;j+m+1jjzj × j=1 BOUNDED l-INDEX AND l-M-INDEX AND COMPOSITIONS OFANALYTIC FUNCTIONS183 n ( ) o (j) q1 q2 × max f + + ::: + qp+1 : 0 ≤ j ≤ m ≤ n zp( zp−1 ) o q q ≤ C max f (k) 1 + 2 + ::: + q : 0 ≤ k ≤ m 1 zp zp−1 p+1 for all jzj ≤ 1: Using (7), we deduce ( ) q q 1 (k) 1 2 ≤ j (k) jj j(p+1)k f p + p−1 + ::: + qp+1 F (z) z p−1 k + z z jq1p + q2(p − 1)z + ::: + qpz j − Xk 1 jQ (z)jjzjp(k−j) + jF (j)(z)jjzj(p+1)j j;k ≤ jq p + q (p − 1)z + ::: + q zp−1j2k−1 j=1 1 2 p ! − Xk 1 jQ (z)jjzjp(k−j) { } ≤ j;k j (j) jj j(p+1)j ≤ ≤ ≤ 1 + p−1 2k−1 max F (z) z : 1 j k jq1p + q2(p − 1)z + ::: + qpz j j=1 { } ≤ 2 max jF (j)(z)jjzj(p+1)j : 1 ≤ j ≤ k for all jzj < "; where " > 0 is sufficiently small and such that the disc jzj ≤ " does not p−1 contain zeros of the polynomial q1p + q2(p − 1)z + ::: + qpz : It is possible because q1 =6 0: Therefore, { } (m+1) (p+1)(m+1) (j) (p+1)j jF (z)jjzj ≤ 2C1 max jF (z)jjzj : 1 ≤ j ≤ mk : Using Theorem 1 and 2, we conclude that the function F has bounded l-index with l(jzj) = β jzjp+1 : j j β On the contrary, let F be of bounded l-index with l( z ) = jzjp+1 : As above, by Theorem 1 from (7) we obtain ( ) q q { } f (m+1) 1 + 2 + ::: + q ≤ C max jF (j)(z)jjzj(p+1)j : 0 ≤ j ≤ m ; zp zp−1 p+1 1 and (6) yields n ( ) o q q jF (j)(z)jjzj(p+1)j ≤ 2 max jf (k) 1 + 2 + ::: + q j: 0 ≤ k ≤ j zp zp−1 p+1 for all jzj < "; where " > 0 is sufficiently small.