# Value distribution of L-functions concerning shared values and certain differential polynomials.

1. Introduction and main results. L-functions, with the Riemann zeta function as a prototype, are important objects in number theory, and value distribution of L-functions has been studied extensively, which can be found, for example in Steuding [12]. Value distribution of L-functions concerns distribution of zeros of L-functions L and, more generally, the c-points of L, i.e., the roots of the equation L(s) = c, or the points in the pre-image [L.sup.-1] = {s [member of] C : L(s) = c}, where and in what follows, s denotes a complex variable in the complex plane C and c denotes a value in the extended complex plane C [union] {[infinity]}. L-functions can be analytically continued as meromorphic functions in C. It is well-known that a nonconstant meromorphic function in C is completely determined by five such pre-images (cf. [2,10,15,17]), which is a famous theorem due to Nevanlinna and often referred to as Nevanlinna's uniqueness theorem. Two meromorphic functions f and g in the complex plane are said to share a value c [member of] C [union] {[infinity]} IM (ignoring multiplicities) if [f.sup.-1] (c) = [g.sup.-1] (c) as two sets in C. Moreover, f and g are said to share a value c CM (counting multiplicities) if they share the value c and if the roots of the equations f (s) = c and g(s) = c have the same multiplicities. Throughout the paper, an L-function always means an L-function L in the Selberg class, which includes the Riemann zeta function [zeta](s) = [[infinity].summation over (n=1)] [n.sup.-s] and essentially those Dirichlet series where one might expect a Riemann hypothesis. Such an L-function is defined to be a Dirichlet series L(s) = [[infinity].summation over (n=1)] a(n)[n.sup.-s] satisfying the following axioms (cf. [11, 12]): (i) Ramanujan hypothesis. a(n) [less than or equal to] [n.sup.[epsilon]] for every [epsilon] > 0. (ii) Analytic continuation. There is a nonnegative integer k such that [(s - 1).sup.k] L(s) is an entire function of finite order. (iii) Functional equation. L satisfies a functional equation of type [[LAMBDA].sub.L] (s) = w[[LAMBDA].sub.L] (1 - s), where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with positive real numbers Q, [[LAMBDA].sub.j] and complex numbers [v.sub.j], w with [Rev.sub.j] [greater than or equal to] 0 and [absolute value of (w)] = 1. (iv) Euler product hypothesis. L(s) = [[PI].sub.p] exp ([[infinity].summation over (k=1)] b([p.sup.k])/[p.sup.ks]) with suitable coefficients b([p.sup.k]) satisfying b([p.sup.k]) [less than or equal to] [p.sup.k[theta]] for some [theta] < 1/2, where the product is taken over all prime numbers p.

We first recall the following result due to Steuding [12], which actually holds without the Euler product hypothesis:

Theorem A ([12, p. 152]). If two L-functions [L.sub.1] and [L.sub.2] with a(1) = 1 share a complex value c [nit equal to] [infinity] CM, then [L.sub.1] = [L.sub.2].

Later on, Li [7] proved the following result to deal with a question posed by Chung-Chun Yang (cf. [7]):

Theorem B ([7]). Let a and b be two distinct finite values, and let f be a meromorphic function in the complex plane such that f has finitely many poles in the complex plane. If f and a nonconstant L-function L share a CM and b IM, then L = f.

In 1997, Lahiri [4] posed the following question: What can be said about the relationship between two meromorphic functions f and g, when two differential polynomials, generated by f and g respectively, share some nonzero finite value? In this direction, Fang [1] and Yang-Hua [14] respectively proved the following results:

Theorem C ([1]). Let f and g be two non-constant entire functions, and let n and k be two positive integers such that n > 2k + 4. If [([f.sup.n]).sup.(k)] and [([g.sup.n]).sup.(k)] share 1 CM, then either f (z) = [c.sub.l][e.sup.cz], g(z) = [c.sub.2][e.sup.-cz], where [C.sub.1], [C.sub.2] and c are three constants satisfying [(-l).sup.k] [([c.sub.1] [c.sub.2]).sup.n] [(nc).sup.2k] = 1, or f = tg for a constant t such that [t.sup.n] = 1.

Theorem D ([14]). Let f and g be two nonconstant meromorphic functions, and let n > 11 be a positive integer. If [f.sup.n] f' and [g.sup.n] g' share 1 CM, then either f (z) = [c.sub.l][e.sup.cz], g(z) = [c.sub.2][e.sup.-cz], where [C.sub.1], [C.sub.2] and c are three constants satisfying [([c.sub.1] [c.sub.2]).sub.n+1] [c.sup.2] = - 1, or f = tg for a constant t such that [t.sup.n+1] = 1.

Regarding Theorems A D, one may ask, what can be said about the relationship between a meromorphic function f and an L-function L, if [(fn).sup.(k)] and [(Ln).sup.(k)] share 1 CM or that [(fn).sup.(k)] and [(Ln).sup.(k)] have the same fixed points, where n and k are positive integers? In this direction, we will prove the following two results respectively:

Theorem 1.1. Let f be a nonconstant meromorphic function, let L be an L-function, and let n and k be two positive integers with n > 3k + 6. If [(fn).sup.(k)] and [(Ln).sup.(k)] share 1 CM, then f = tL for a constant t satisfying [t.sup.n] = 1.

Theorem 1.2. Let f be a nonconstant meromorphic function, let L be an L-function, and let n and k be two positive integers with n> 3k + 6. If [(fn).sup.(k)] (z) - z and [(Ln).sup.(k)] (z) - z share 0 CM, then f = tL for a constant t satisfying [t.sup.n] = 1.

To prove Theorems 1.1 and 1.2 in the present paper, we will apply Nevanlinna theory, which can be found in [2,6,15,17]. In addition, we will use the lower order [mu](f) and the order [rho](f) of a meromorphic function f, which can be found, for example in [2,6,17], are in turn defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We also need the following two definitions:

Definition 1.1 ([5, Definition 1]). Let p be a positive integer and a [member of] C [union]{[infinity]}. Next we denote by [N.sub.p]) (r, 1/f-a) the counting function of those a-points of f (counted with proper multiplicities) whose multiplicities are not greater than p, and denote by [N.sub.(p] (r, 1/f-a) the counting function of those a-points of f (counted with proper multiplicities) whose multiplicities are not less than p. We denote by [[bar.N].sub.p)] (r , 1/f-a) and [[bar.N].sub.p)] (r, 1/f-a) the reduced forms of [[bar.N].sub.p)] (r, 1/f-a) and [[bar.N].sub.p)] (r, 1/f-a) respectively.

Here [N.sub.p)] (r, 1/f-[infinity]), [[bar.N].sub.p)] (r, 1/f-[infinity]), [[bar.N].sub.p)] (r, 1/f-[infinity]) and [[bar.N].sub.p)] (r, 1/f-[infinity]) mean [N.sub.p)] (r, f), [[bar.N].sub.p)] (r,f), [N.sub.p)] (r,f) and [[bar.N].sub.p)] (r, f) respectively.

Definition 1.2. Let a be an any value in the extended complex plane and let k be an arbitrary nonnegative integer. We define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[N.sub.k] (r, 1/f-a) = [bar.N](r, 1/f-a) + [[bar.N].sub.2] it(r, 1/f-a)

+ ... + [[bar.N].sub.(k](r, 1/f-a).

Remark 1.1. By Definition 1.2 we have

0 [less than or equal to] [[delta].sub.k] (a, f) [less than or equal to] [[delta].sub.k-1] (a, f) [less than or equal to] [[delta].sub.1] (a,f) [less than or equal to] [THETA](a,f) [less than or equal to] 1.

2. Preliminaries. In this section, we will give the following lemmas that play an important role in proving the main results in this paper:

Lemma 2.1 ([2, Theorem 3.2] and [17, Theorem 4.3]). Let f be a nonconstant meromorphic function, let k [greater than or equal to] 1 be a positive integer, and let c be a nonzero finite complex number. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [N.sub.0] (r, 1/[f.sup.(k+1)])) is the counting function of those zeros of [f.sup.(k+1)] in [absolute value of (z)] < r which are not zeros of f([f.sup.(k)] - c) in [absolute value of (z)] < r.

Lemma 2.2 ([8, Lemma 2.5]). Let F and G be two nonconstant meromorphic functions such that [F.sup.(k)] - P and [G.sup.(k)] - P share 0 CM, where k [greater than or equal to] 1 is a positive integer, P [??] 0 is a polynomial. If

(k + 2)[THETA]([infity], F) + 2[THETA]([infity], G) + [THETA](0, F) + [THETA](0, G)

+ [[delta].sub.k+1] (0, F)+ [[delta].sub.k+1] (0,G) > k + 7

and

(k + 2)[THETA]([infinity], G) + 2[THETA]([infinity], F) + [THETA] (0, G) + [THETA] (0, F)

+ [[delta].sub.k+1] (0, G) + [[delta].sub.k+1] (0,F) > k + 7,

then either [F.sup.(k)] [G.sup.(k)] = [P.sup.2] or F = G.

Lemma 2.3 ([15, Theorem 1.24]). Suppose that f is a nonconstant meromorphic function in the complex plane and k is a positive integer. Then

N (r, 1/[f.sup.(k)])) [less than or equal to] N (r, 1/f) + k[bar.N] (r, f)

+ O(log T (r,f) + log r),

as r [right arrow] [infinity], outside of a possible exceptional set of finite linear measure.

Lemma 2.4 ([18, Lemma 6]). Let [f.sub.1] and [f.sub.2] be two nonconstant meromorphic functions satisfying [bar.N](r,[f.sub.j]) + [bar.N] (r, 1/[f.sub.j]) = S(r), (j = 1, 2). Then, either [[bar.N].sub.0] (r, 1; [f.sub.1], [f.sub.2]) = S(r) or there exist two integers p and q satisfying [absolute value of (p)] + [absolute value of (q)] > 0, such that [f.sup.p.sub.1] [f.sup.q.sub.2] = 1, where [[bar.N].sub.0] (r, 1; [f.sub.1],[f.sub.2]) denotes the reduced counting function of the common 1-points of [f.sub.1] and [f.sub.2] in [absolute value of (z)] < r, T(r) = T(r, [f.sub.1]) + T(r, [f.sub.2]) and S(r) = o(T(r)), as r [??]E and r [right arrow] [infinity]. Here E [subset] (0, + [infinity]) is a subset of finite linear measure.

Lemma 2.5 ([3]). Let f be a transcendental meromorphic function in C. Then, for each K > 1, there exists a set M(K) [subset] (0, + [infinity]) of the lower logarithmic density at most d(K) = 1 - [(2[e.sup.K-1] - 1).sup.-1] > 0, that is

, 1 f dt

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that, for every positive integer k,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Lemma 2.6 ([16, proof of Lemma 1]). Let f be a nonconstant meromorphic function, let k [greater than or equal to] 1 be a positive integer, and let [phi] [??] 0, [infinity] be a small function of f, i.e., T(r,[phi]) = S(r, f). Then

T(r, f) [less than or equal to] [bar.N](r, f) + N (r, 1/f) + N (r, 1/[f.sup.(k)] - [phi])

- N (r, 1/[f.sup.(k)/[phi]]') + S(r,f).

3. Proof of Theorems 1.1 and 1.2.

Proof of Theorem 1.1. First of all, we denote by d the degree of L. Then d = 2 [K.summation over (j=1)] [[lambda].sub.j] > 0 (cf. [12, p. 113]), where K and [[lambda].sub.j] are respectively the positive integer and the positive real number in the functional equation of the axiom (iii) of the definition of L-functions. Therefore, by Steuding [12, p. 150] we have

(3.1) T(r, L) = d/[pi] r log r + O(r).

Noting that an L-function at most has one pole z = 1 in the complex plane, we deduce by Lemmas 2.1 and Valiron-Mokhonko lemma (cf. [9]) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(3.2) (n - k - 1)T(r, L) [less than or equal to] T(r, [([f.sup.n]).sup.(k)]) + O(log r).

By (3.1) we see that L is a transcendental meromorphic function. Combining this with (3.2), Theorem 1.5 [15] and the assumption n> 3k + 6, we deduce that [([f.sup.n]).sup.(k)], and so f is a transcendental meromorphic function. Now we let

(3.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

(3.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By Valiron-Mokhonko lemma we have

(3.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

(3.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Noting that an L-function at most has one pole z = 1 in the complex plane, we have by (3.1) that

(3.8) [THETA]([infinity], [L.sup.n]) = 1.

By (3.3), (3.5) (3.8) we have

(3.9) [[DELTA].sub.1] > k + 8 - 3k + 6/n, [[DELTA].sub.2] > k + 8- 2k + 6/n.

By (3.9) and the assumption n > 3k + 6 we have [[DELTA].sub.1] > k + 7 and [[DELTA].sub.2] > k + 7. This together with (3.3), (3.4) and Lemma 2.2 gives [([f.sup.n]).sup.(k)] [([L.sup.n]).sup.(k)] = 1 or [f.sup.n] = [L.sup.n]. We consider the following two cases:

Case 1. Suppose that [([f.sup.n]).sup.(k)] [([L.sup.n]).sup.(k)] = 1. First of all, we prove that 0 is a Picard exceptional value of f and L. Indeed, suppose that [z.sub.0] [member of] C is a zero of f with multiplicity m. Then, by the assumption [([f.sup.n]).sup.(k)] [([L.sup.n]).sup.(k)] = 1 we can find that [z.sub.0] = 1 is a pole of L with multiplicity, say p, such that mn - k = np + k, and so (m - p)n = 2k, and so we have n [less than or equal to] 2k, which contradicts the assumption n > 3k + 6. Similarly, we can prove that 0 is a Picard exceptional value of L. On the other hand, by (3.1), Valiron-Mokhonko lemma, the assumption [([f.sup.n]).sup.(k)] [([L.sup.n]).sup.(k)] = 1, a result from Whittaker [13, p. 82] and the definition of the order of a meromorphic function we have

(3.10) [rho] = [rho] ([f.sup.n]) = p([([f.sup.n]).sup.(k)]) = p([([L.sup.n]).sup.(k)])

= [rho]([L.sup.n]) = [rho](L) = 1.

Noting that L has at most one pole z = 1 in the complex plane, we have by (3.10), Lemma 2.3 and [([f.sup.n]).sup.(k)] [([L.sup.n]).sup.(k)] = 1 that

(3.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore,

(3.12) [bar.N](r, f) + [bar.N] (r, L) [less than or equal to] O(log r).

Now we set

(3.13) [f.sub.1] = [([f.sup.n]).sup.(k)]/[([L.sup.n]).sup.(k)], [f.sub.2] = [([f.sup.n]).sup.(k)] - 1/[([L.sup.n]).sup.(k)] - 1.

By (3.13) and the assumption that f and L are transcendental meromorphic functions, we have [f.sub.1] [??] 0 and [f.sub.2] [??] 0. Suppose that one of [f.sub.1] and [f.sub.2] is a nonzero constant. Then, by (3.13) we see that [([f.sup.n]).sup.(k)] and [([L.sup.n]).sup.(k)] share [infinity] CM. Combining this with [([f.sup.n]).sup.(k)] [([L.sup.n]).sup.(k)] = 1 we deduce that [infinity] is a Picard exceptional value of f and L. Next we suppose that [f.sub.1] and [f.sub.2] are nonconstant meromorphic functions. We set

(3.14) [F.sub.1] = [([f.sup.n]).sup.(k)], [G.sub.1] = [([L.sup.n]).sup.(k)].

Then, by (3.13) and (3.14) we have

(3.15) [F.sub.1] = [f.sub.1](1 - [f.sub.2])/[f.sub.1] - [f.sub.2], [G.sub.1] = 1 - [f.sub.2])/[f.sub.1] -

[f.sub.2].

By (3.15) we can find that there exists a subset I [subset] (0, + [infinity]) with infinite linear measure such that S(r) = o(T(r)) and

(3.16) T(r, [F.sub.1]) [less than or equal to] 2(T(r, [f.sub.1]) + T(r, [f.sub.2])) + S(r)

[less than or equal to] 8T(r,[F.sub.1]) + S(r)

or

(3.17) T(r, [G.sub.1]) [less than or equal to] 2(T(r, [f.sub.1]) + T(r, [f.sub.2])) + S(r)

[less than or equal to] 8T(r,[G.sub.1]) + S(r)

as r [member of] I and r [right arrow] [infinity], where T(r) = T(r, [f.sub.1]) + T(r, [f.sub.2]). Without loss of generality, we suppose that (3.16) holds. Then we have S(r) = S(r, [F.sub.1]), as r [member of] I and r [right arrow] [infinity]. By [([f.sup.n]).sup.(k)] [([L.sup.n]).sup.(k)] = 1 we see that [([f.sup.n]).sup.(k)] and [([L.sup.n]).sup.(k)] share 1 and -1 CM. Noting that 0 is a Picard exceptional value of f and L, we deduce by (3.10) and Lemma 2.3 that

(3.18) N (r, 1/[([f.sup.n]).sup.(k)]) [less than or equal to] k [bar.N] (r, f) + O (log r).

By (3.11), (3.12) and (3.18) we have

(3.19) N (r, 1/[([f.sup.n]).sup.(k)]) + N (r, 1/[([L.sup.n]).sup.(k)]) [less than or equal to] O(log r).

Noting that [([f.sup.n]).sup.(k)] and [([L.sup.n]).sup.(k)] are transcendental meromorphic functions such that [([f.sup.n]).sup.(k)] and [([L.sup.n]).sup.(k)] share 1 CM, we deduce by (3.12), (3.13) and (3.19) that

(3.20) [bar.N](r, 1/[f.sub.j]) + [bar.N](r, [f.sub.j]) = o(T(r)), (j = 1, 2),

as r [member of] I and r [right arrow] [infinity]. Noting that [([f.sup.n]).sup.(k)] and [([L.sup.n]).sup.(k)] share -1 CM, we deduce by (3.12), (3.14), (3.16), (3.18) and the second fundamental theorem that

(3.21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

as r [member of] I and r [right arrow] [infinity]. By (3.16) and (3.21) we have

(3.22) T(r, [f.sub.i]) + T(r, [f.sub.2]) [less than or equal to] [[bar.N].sub.0] (r, 1; [f.sub.1], [f.sub.2]) + o(T(r)).

By (3.13), (3.14), (3.20), (3.22) and Lemma 2.4 we find that there exist two relatively prime integers s and t satisfying [absolute value of (s)] + [absolute value of (t)] > 0, such that [f.sup.s.sub.1] [f.sup.t.sub.2] = 1. Combining this with (3.13) and (3.14), we have

(3.23) [([F.sub.1]/[G.sub.1]).sup.s] [([F.sub.1] - 1/[G.sub.1] - 1).sup.t] = 1.

By (3.23) we consider the following two subcases:

Subcase 1.1. Suppose that st < 0, say s > 0 and t < 0, say t = - [t.sub.1], where [t.sub.1] is some positive integer. Then, (3.23) can be rewritten as

(3.24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [z.sub.1] [member of] C be a pole of [F.sub.1] of multiplicity [p.sub.1] [greater than or equal to] 1. Then, by [F.sub.1] [G.sub.1] = 1 we can see that [z.sub.1] is a zero of [G.sub.1] of multiplicity [p.sub.1]. Therefore, by (3.24) we deduce that 2s = [t.sub.1] = - t. Combining this with the assumption that s and t are two relatively prime integers, we have s = 1 and t = -[t.sub.1] = -2. Therefore, (3.24) can be rewritten as [F.sub.1] [([G.sub.1] - 1).sup.2] = [G.sub.1] [([F.sub.1] - 1).sup.2], this is equivalent to the obtained result [F.sub.1] [G.sub.1] =1i. Next we can deduce a contradiction by using the other method. Indeed, by (3.19) and the fact that L, and so [L.sup.(k)] have at most one pole z = 1 in the complex plane, we have

(3.25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [P.sub.1] is a nonzero polynomial, [p.sub.2] [greater than or equal to] 0 is an integer, [A.sub.1] [not equal to] 0 and [B.sub.1] are constants. By (3.25), Lemma 2.5 and Hayman [2,p. 7] we deduce that there exists a subset I [subset] (0, + [infinity]) with logarithmic measure logmeas I = [[integral].sub.I] dt/t = [infinity] such that for some given sufficiently large positive number K > 1, we have

(3.26) T(r, L) < 3eKT(r, [([L.sup.n]).sup.(k)]

= 3eK [absolute value of ([A.sub.1])]r/[pi] (i + o(1)) + 0(log r),

as r [member of] I and r [right arrow] [infinity]. By (3.1) and (3.26) we have a contradiction.

Subcase 1.2. Suppose that st = 0 or st > 0. Then, by (3.23) we can see that [F.sub.1] and [G.sub.1] share [infinity] CM. This together with (3.14) and the assumption [([f.sup.n]).sup.(k)] [([L.sup.n]).sup.(k)] = 1 implies that [infinity] is a Picard exceptional value of f and L: Combining this with the obtained result that 0 is a Picard exceptional value of f and L, we have

(3.27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [A.sub.2] [not equal to] 0 and [B.sub.2] are constants. By (3.27) and Hayman [2, p. 7] we have

(3.28) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Case 2. Suppose that [f.sup.n] = [L.sup.n]. Then, we have f = tL, where t is a constant satisfying [t.sup.n] = 1. This completes the proof of Theorem 1.1.

Proof of Theorem 1.2. First of all, in the same manner as in the beginning of the proof of Theorem 1.1 we have (3.1). Now we let [z.sub.2] [member of] C be a zero of L with multiplicity [p.sub.2]. Then [z.sub.2] is a zero of [L.sup.n] with multiplicity n[p.sub.2], and so [z.sub.2] is a zero of ([([L.sup.n]).sup.(k)]/z)' with multiplicity n[p.sub.2] - k - 2 at least. Again let [z.sub.3] be a zero of [([L.sup.n]).sup.(k)]/z - 1 with multiplicity [p.sub.3]. Then, [z.sub.3] is a zero of ([([L.sup.n]).sup.(k)]/z)' with multiplicity [p.sub.3] - 1. Then, by (3.1), Lemma 2.6 and the value sharing assumption we have

(3.29) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where No (r, 1/([([L.sup.n]).sup.(k)]/z)') is the counting function of those zeros of ([([L.sup.n]).sup.(k)]/z)' in [absolute value of (z)] < r that are not zeros of [([L.sup.n]).sup.(k)]/z in [absolute value of (z)] < r. By Valiron-Mokhonko lemma we have T(r, [L.sup.n]) = nT(r, L)+ O(1). This together with

(3.29) gives

(3.30) (n - k - 2)T(r, L) [less than or equal to] T(r, ([[f.sup.n]).sup.(k)]) + O(log r).

By (3.30) and the assumption n> 3k + 6, we deduce that [([f.sup.n]).sup.(k)], and so f is a transcendental meromorphic function. Next in the same manner as in the proof of Theorem 1.1 we have [([f.sup.n]).sup.(k)] [([L.sup.n]).sup.(k)] = [z.sup.2] or [f.sup.n] = [L.sup.n] by Lemma 2.2. We consider the following two cases:

Case 1. Suppose that [([f.sup.n]).sup.(k)] [([L.sup.n]).sup.(k)] = [z.sup.2].

Then, [F.sub.2][G.sub.2] = l, where

(3.31) [F.sub.2] = [([f.sup.n]).sup.(k)]/z, [G.sub.2] = [([g.sup.n]).sup.(k)]/z.

Next, in the same manner as in Case 1 of the proof of Theorem 1.1 we can get a contradiction.

Case 2. Suppose that [f.sup.n] = [L.sup.n]. Then we get the conclusion of Theorem 1.2. This completes the proof of Theorem 1.2.

doi:10.3792/pjaa.93.41

Acknowledgments. The authors want to express their thanks to the referee for his/her valuable suggestions and comments concerning this paper. This work is supported in part by the NSFC (No. 11171184), the NSFC (No. 11461042) and the NSF of Shandong Province, China (No. ZR2014AM011).

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Fang LIU, *) Xiao-Min LI *) and Hong-Xun YI **) (Communicated by Shigefumi MORI, M.J.A., April 12, 2017)

2010 Mathematics Subject Classification. Primary 11M36; Secondary 30D35.

*) Department of Mathematics, Ocean University of China, Qingdao, Shandong 266100, P. R. China.

**) Department of Mathematics, Shandong University, Jinan, Shandong 250199, P. R. China.