International Journal of Analysis and Applications ISSN 2291-8639 Volume 5, Number 1 (2014), 81-90 http://www.etamaths.com GROWTH PROPERTIES OF WRONSKIANS IN THE LIGHT OF RELATIVE ORDER SANJIB KUMAR DATTA1,∗, TANMAY BISWAS2, GOLOK KUMAR MONDAL3 Abstract. In this paper we study the comparative growth properties of com- position of entire and meromorphic functions on the basis of relative order (relative lower order) of Wronskians generated by entire and meromorphic functions. 1. Introduction, Definitions and Notations. Let f be an entire function defined in the open complex plane C. The func- tion M (r,f) on |z| = r known as maximum modulus function corresponding to f is defined as follows: M(r,f) = max|z|=r |f (z)| . When f is meromorphic, M (r,f) can not be defined. In this situation one may define another function T (r,f) known as Nevanlinna’s Characteristic function of f, playing the same role as M (r,f) in the following manner: T (r,f) = N (r,f) + m (r,f) . When f is an entire function, T (r,f) reduces to m (r,f) . We call the function N (r,a; f) ( − N (r,a; f) ) as counting function of a-points (distinct a-points) of f. In many occasions N (r,∞; f) and − N (r,∞; f) are denoted by N (r,f) and − N (r,f) respectively.We put N (r,a; f) = r∫ 0 n (t,a; f) −n (0,a; f) t dt + − n (0,a; f) log r , where we denote by n (t,a; f) ( − n (t,a; f) ) the number of a-points (distinct a-points) of f in |z| ≤ t and an ∞ -point is a pole of f . 2010 Mathematics Subject Classification. 30D20, 30D30, 30D35. Key words and phrases. Entire function; meromorphic function; relative order (relative lower order); Wronskian. c©2014 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 81 82 DATTA, BISWAS ABD MONDAL On the other hand m ( r, 1 f−a ) is denoted by m (r,a; f) and we mean m (r,∞; f) by m (r,f) , which is called the proximity function of f. We also put m (r,f) = 1 2π 2π∫ 0 log+ ∣∣f (reiθ)∣∣dθ, where log+ x = max (log x, 0) for all x > 0 . If the entire function g is non-constant then Tg (r) is strictly increasing and continuous and its inverse T−1g : (Tg (0) ,∞) → (0,∞) exists and is such that lim s→∞ T−1g (s) = ∞. Lahiri and Banerjee [4] introduced the definition of relative order of a mero- morphic function with respect to an entire function which is as follows: Definition 1.1. [4] Let f be meromorphic and g be entire. The relative order of f with respect to g denoted by ρg (f) is defined as ρg (f) = inf {µ > 0 : Tf (r) < Tg (rµ) for all sufficiently large r} = lim sup r→∞ log T−1g Tf (r) log r . Analogously, one can define the relative lower order of a meromorphic func- tion f with respect to an entire function g denoted by λg (f) in the following manner : λg (f) = lim inf r→∞ log T−1g Tf (r) log r . If we consider g (z) = exp z, Definition 1.1 coincides {cf.[4]} with the classi- cal definition of order and lower order of meromorphic function which are as follows: Definition 1.2. The order ρf and lower order λf of a meromorphic function f are defined as ρf = lim sup r→∞ log Tf (r) log r and λf = lim inf r→∞ log Tf (r) log r where log[k] x = log ( log[k−1] x ) , k = 1, 2, 3, ... and log[0] x = x. The following definitions are also well known: Definition 1.3. A meromorphic function a ≡ a (z) is called small with respect to f if T (r,a) = S (r,f) where S (r,f) = o{T (r,f)} i.e., S(r,f) T(r,f) → 0 as r →∞ . Definition 1.4. Let a1,a2, ....ak be linearly independent meromorphic functions and small with respect to f .We denote by L (f) = W (a1,a2, ....ak; f) , the Wron- skian determinant of a1,a2, ....,ak, f i.e., L (f) = ∣∣∣∣∣∣∣∣∣∣∣∣ a1 a2 . . . ak f a ′ 1 a ′ 2 . . . a ′ k f ′ . . . . . . . . . . . . . . . . . . . . . a (k) 1 a (k) 2 . . . a (k) k f (k) ∣∣∣∣∣∣∣∣∣∣∣∣ . GROWTH PROPERTIES OF WRONSKIANS IN THE LIGHT OF RELATIVE ORDER 83 Definition 1.5. If a ∈ C∪{∞},the quantity δ (a; f) = 1 − lim sup r→∞ N (r,a; f) Tf (r) = lim inf r→∞ m (r,a; f) Tf (r) is called the Nevanlinna’s deficiency of the value “a”. From the second fundamental theorem it follows that the set of values of a ∈ C ∪ {∞} for which δ (a; f) > 0 is countable and ∑ a6=∞ δ (a; f) + δ (∞; f) ≤ 2 (cf [3],.p.43 ). If in particular, ∑ a 6=∞ δ (a; f) + δ (∞; f) = 2, we say that f has the maximum deficiency sum. In this connection the following two definitions are also relevant : Definition 1.6. [1] A non-constant entire function f is said have the property (A) if for any δ > 1 and for all large r, [Mf (r)] 2 ≤ Mf ( rδ ) holds. For exapmles of functions with or without the Property (A), one may see [1]. Definition 1.7. Two entire functions g and h are said to be asymptotically equiv- alent if there exists l (0 < l < ∞) such that Mg (r) Mh (r) → l as r →∞ and in this case we write g ∼ h . Clearly if g ∼ h then h ∼ g. In this paper we establish some newly developed results based on the growth properties of relative order and relative lower order of wronskians generated by entire and meromorphic functions. We do not explain the standard notations and definitions in the theory of entire and meromorphic functions because those are available in [3] and [5]. 2. Lemmas. In this section we present some lemmas which will be needed in the sequel. Lemma 2.1. [1] Let g be an entire function and α > 1, 0 < β < α. Then Mg (αr) > βMg (r) for all sufficiently large r. Lemma 2.2. [1] Let f be an entire function which satisfies the Property (A). Then for any positive integer n and for all large r, [Mf (r)] n ≤ Mf ( rδ ) holds where δ > 1. Lemma 2.3. Let g be entire. Then for all sufficiently large values of r, Tg (r) ≤ log Mg (r) ≤ 3Tg (2r) . Lemma 3 follows from Theorem 1.6 {cf. p.18, [3]} on putting R = 2r. Lemma 2.4. [2] If f be a transcendental meromorphic function with the maximum deficiency sum and g be a transcendental entire function of regular growth having non zero finite order and ∑ a6=∞ δ (a; g) + δ (∞; g) = 2, then the relative order and relative lower order of L(f) with respect to L(g) are same as those of f with respect to g i.e., ρL[g] (L [f]) = ρg (f) and λL[g] (L [f]) = λg (f) . 84 DATTA, BISWAS ABD MONDAL Lemma 2.5. Let g and h be any two transcendental entire functions of regular growth having non zero finite order with ∑ a6=∞ δ (a; g)+δ (∞; g) = 2 and ∑ a6=∞ δ (a; h)+ δ (∞; h) = 2 respectively. Then for any transcendental meromorphic function f with the maximum deficiency sum, ρL[g] (L[f]) = ρL[h] (L[f]) and λL[g] (L[f]) = λL[h] (L[f]) , if g and h have the Property (A) and g ∼ h. Proof. Let ε > 0 be arbitrary. Now we get from Definition 1.7 and Lemma 2.1 for all sufficiently large values of r that (1) Mg (r) < (l + ε) Mh (r) ≤ Mh (αr) , where α > 1 is such that l + ε < α. Now from Lemma 2.3 and in view of Definition 1.1, we obtain for all sufficiently large values of r that Tf (r) ≤ Tg [ (r) (ρg(f)+ε) ] i.e., Tf (r) ≤ log Mg [ (r) (ρg(f)+ε) ] . Therefore in view of (1), for any δ > 1 it follows from above by using Lemma 2.2 and Lemma 2.3 that Tf (r) ≤ 1 3 log [ Mh [ (αr) (ρg(f)+ε) ]]3 i.e., Tf (r) ≤ 1 3 log Mh [ (αr) δ(ρg(f)+ε) ] i.e., Tf (r) ≤ Th [ (2αr) δ(ρg(f)+ε) ] i.e., log T−1h Tf (r) log r ≤ δ (ρg (f) + ε) log (2αr) log r . Letting δ → 1+ we get from above that (2) ρh (f) ≤ ρg (f) . Since h ∼ g, we also obtain that (3) ρg (f) ≤ ρh (f) . Now in view of Lemma 2.4 we obtain from (2) and (3) that ρL[g] (L[f]) = ρL[h] (L[f]) . Similarly we have λL[g] (L[f]) = λL[h] (L[f]) . Thus the lemma follows. � GROWTH PROPERTIES OF WRONSKIANS IN THE LIGHT OF RELATIVE ORDER 85 3. Theorems. In this section we present the main results of the paper. Theorem 3.1. Suppose f be a transcendental meromorphic function having the maximum deficiency sum. Also let h be a transcendental entire function of regular growth having non zero finite order with ∑ a6=∞ δ (a; h) + δ (∞; h) = 2 and g be any entire function such that 0 < λh (f ◦g) ≤ ρh (f ◦g) < ∞ and 0 < λh (f) ≤ ρh (f) < ∞ .Then λh (f ◦g) ρh (f) ≤ lim inf r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) ≤ λh (f ◦g) λh (f) ≤ lim sup r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) ≤ ρh (f ◦g) λh (f) . Proof. From the definition of ρh (f) and λh (f ◦g) and Lemma 2.4 we have for arbitrary positive ε and for all sufficiently arge values of r that (4) log T−1h Tf◦g (r) > (λh (f ◦g) −ε) log r and log T−1 L[h] TL[f] (r) ≤ ( ρL[h] (L[f]) + ε ) log r i.e., log T−1 L[h] TL[f] (r) ≤ (ρh (f) + ε) log r .(5) Now from (4) and (5) it follows for all sufficiently large values of r that log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) > (λh (f ◦g) −ε) log r (ρh (f) + ε) log r . As ε (> 0) is arbitrary , we obtain that (6) lim inf r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) > λh (f ◦g) ρh (f) . Again for a sequence of values of r tending to infinity , (7) log T−1h Tf◦g (r) ≤ (λh (f ◦g) + ε) log r and for all sufficiently large values of r , log T−1 L[h] TL[f] (r) > ( λL[h] (L[f]) −ε ) log r i.e., log T−1 L[h] TL[f] (r) > (λh (f) −ε) log r .(8) Combining (7) and (8) we get for a sequence of values of r tending to infinity that log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) ≤ (λh (f ◦g) + ε) log r (λh (f) −ε) log r . Since ε (> 0) is arbitrary, it follows that (9) lim inf r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) ≤ λh (f ◦g) λh (f) . 86 DATTA, BISWAS ABD MONDAL Also for a sequence of values of r tending to infinity, log T−1 L[h] TL[f] (r) ≤ ( λL[h] (L[f]) + ε ) log r i.e., log T−1 L[h] TL[f] (r) ≤ (λh (f) + ε) log r .(10) Now from (4) and (10) we obtain for a sequence of values of r tending to infinity that log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) ≥ (λh (f ◦g) −ε) log r (λh (f) + ε) log r . As ε (> 0) is arbitrary, we get from above that (11) lim sup r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) ≥ λh (f ◦g) λh (f) . Also for all sufficiently large values of r , (12) log T−1h Tf◦g (r) ≤ (ρh (f ◦g) + ε) log r . Now it follows from (8) and (12) for all sufficiently large values of r that log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) ≤ (ρh (f ◦g) + ε) log r (λh (f) −ε) log r . Since ε (> 0) is arbitrary, we obtain that (13) lim sup r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) ≤ ρh (f ◦g) λh (f) . Thus the theorem follows from (6) , (9) , (11) and (13) . � The following theorem can be proved in the line of Theorem 3.1 and so its proof is omitted. Theorem 3.2. Let g be a transcendental entire function with ∑ a6=∞ δ (a; g)+δ (∞; g) = 2. Also let h be a transcendental entire function of regular growth having non zero finite order with ∑ a6=∞ δ (a; h) + δ (∞; h) = 2 and f be any meromorphic function such that 0 < λh (f ◦g) ≤ ρh (f ◦g) < ∞ and 0 < λh (g) ≤ ρh (g) < ∞ .Then λh (f ◦g) ρh (g) ≤ lim inf r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[g] (r) ≤ λh (f ◦g) λh (g) ≤ lim sup r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[g] (r) ≤ ρh (f ◦g) λh (g) . Theorem 3.3. Suppose f be a transcendental meromorphic function with ∑ a6=∞ δ (a; f) +δ (∞; f) = 2. Also let g be entire and h be a transcendental entire function of regular growth having non zero finite order with ∑ a 6=∞ δ (a; h) + δ (∞; h) = 2, 0 < ρh (f ◦g) < ∞ and 0 < ρh (f) < ∞ .Then lim inf r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) ≤ ρh (f ◦g) ρh (f) ≤ lim sup r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) . GROWTH PROPERTIES OF WRONSKIANS IN THE LIGHT OF RELATIVE ORDER 87 Proof. From the definition of ρL[h] (L[f]) and in view of Lemma 2.4 we get for a sequence of values of r tending to infinity that log T−1 L[h] TL[f] (r) > ( ρL[h] (L[f]) −ε ) log r i.e., log T−1 L[h] TL[f] (r) > (ρh (f) −ε) log r .(14) Now from (12) and (14) it follows for a sequence of values of r tending to infinity that log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) ≤ (ρh (f ◦g) + ε) log r (ρh (f) −ε) log r . As ε (> 0) is arbitrary, we obtain that (15) lim inf r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) ≤ ρh (f ◦g) ρh (f) . Again for a sequence of values of r tending to infinity , (16) log T−1h Tf◦g (r) > (ρh (f ◦g) −ε) log r . So combining (5) and (16) we get for a sequence of values of r tending to infinity that log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) > (ρh (f ◦g) −ε) log r (ρh (f) + ε) log r . Since ε (> 0) is arbitrary, it follows that (17) lim sup r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) > ρh (f ◦g) ρh (f) . Thus the theorem follows from (15) and (17) . � The following theorem can be carried out in the line of Theorem 3.3 and therefore we omit its proof. Theorem 3.4. Let f be meromorphic and g,h be both transcendental entire func- tions with the maximum deficiency sums and 0 < ρh (f ◦g) < ∞ , 0 < ρh (g) < ∞. In addition, let h of regular growth having non zero finite order. Then lim inf r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[g] (r) ≤ ρh (f ◦g) ρh (g) ≤ lim sup r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[g] (r) . The following theorem is a natural consequence of Theorem 3.1 and Theorem 3.3 : Theorem 3.5. Suppose f be a transcendental meromorphic function having the maximum deficiency sum. Also let h be a transcendental entire function of regular growth having non zero finite order with ∑ a6=∞ δ (a; h) + δ (∞; h) = 2 and g be any entire function such that 0 < λh (f ◦g) ≤ ρh (f ◦g) < ∞ and 0 < λh (f) ≤ ρh (f) < ∞ .Then lim inf r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) ≤ min { λh (f ◦g) λh (f) , ρh (f ◦g) ρh (f) } ≤ max { λh (f ◦g) λh (f) , ρh (f ◦g) ρh (f) } ≤ lim sup r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) . 88 DATTA, BISWAS ABD MONDAL The proof is omitted. Analogously one may state the following theorem without its proof. Theorem 3.6. Let f be meromorphic and g,h be both transcendental entire func- tions with the maximum deficiency sums and 0 < λh (f ◦g) ≤ ρh (f ◦g) < ∞ , 0 < λh (g) ≤ ρh (g) < ∞. In addition, let h of regular growth having non zero finite order. Then lim inf r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[g] (r) ≤ min { λh (f ◦g) λh (g) , ρh (f ◦g) ρh (g) } ≤ max { λh (f ◦g) λh (g) , ρh (f ◦g) ρh (g) } ≤ lim sup r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[g] (r) . Theorem 3.7. Suppose f be a transcendental meromorphic function having the maximum deficiency sum. Also let h be a transcendental entire function of regular growth having non zero finite order with ∑ a6=∞ δ (a; h) + δ (∞; h) = 2 and g be any entire function such that ρh (f) < ∞ and λh (f ◦g) = ∞ .Then lim r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[f] (r) = ∞ . Proof. Let us suppose that the conclusion of the theorem do not hold. Then we can find a constant β > 0 such that for a sequence of values of r tending to infinity, (18) log T−1h Tf◦g (r) ≤ β log T −1 L[h] TL[f] (r) . Again from the definition of ρL[g] (L [f]) it follows for all sufficiently large values of r and in view of Lemma 2.4 that log T−1 L[h] TL[f] (r) ≤ ( ρL[h] (L[f]) + ε ) log r i.e., log T−1 L[h] TL[f] (r) ≤ (ρh (f) + ε) log r .(19) Thus from (18) and (19) , we have for a sequence of values of r tending to infinity that log T−1h Tf◦g (r) ≤ β (ρh (f) + ε) log r i.e., log T−1h Tf◦g (r) log r ≤ β (ρh (f) + ε) log r log r i.e., lim inf r→∞ log T−1h Tf◦g (r) log r = λh (f ◦g) < ∞. This is a contradiction. Hence the theorem follows. � Remark 3.8. Theorem 3.7 is also valid with “limit superior” instead of “limit” if λh (f ◦g) = ∞ is replaced by ρh (f ◦g) = ∞ and the other conditions remain the same. Corollary 3.9. Under the assumptions of Theorem 3.7 and Remark 3.8, lim r→∞ T−1h Tf◦g (r) T−1 L[h] TL[f] (r) = ∞ and lim sup r→∞ T−1h Tf◦g (r) T−1 L[h] TL[f] (r) = ∞ . respectively hold. GROWTH PROPERTIES OF WRONSKIANS IN THE LIGHT OF RELATIVE ORDER 89 The proof is omitted. Analogously one may also state the following theorem and corollaries without their proofs as those may be carried out in the line of Remark 3.8, Theorem 3.7 and Corollary 3.9 respectively. Theorem 3.10. Let g be a transcendental entire function with ∑ a6=∞ δ (a; g)+δ (∞; g) = 2. Also let h be a transcendental entire function of regular growth having non zero finite order with ∑ a6=∞ δ (a; h) + δ (∞; h) = 2 and f be any meromorphic function such that ρh (g) < ∞ and ρh (f ◦g) = ∞ .Then lim sup r→∞ log T−1h Tf◦g (r) log T−1 L[h] TL[g] (r) = ∞ . Corollary 3.11. Theorem 3.10 is also valid with “limit” instead of “limit superior” if ρh (f ◦g) = ∞ is replaced by λh (f ◦g) = ∞ and the other conditions remain the same. Corollary 3.12. Under the assumptions of Theorem 3.7 and Corollary 3.11, lim sup r→∞ T−1h Tf◦g (r) T−1 L[h] TL[g] (r) = ∞ and lim r→∞ T−1h Tf◦g (r) T−1 L[h] TL[g] (r) = ∞ respectively hold. Theorem 3.13. Suppose f be a transcendental meromorphic function with ∑ a 6=∞ δ (a; f) +δ (∞; f) = 2. Also let h be a transcendental entire function of regular growth hav- ing non zero finite order with the maximum deficiency sum and g be any entire function such that 0 < ρh (f ◦g) < ∞ and 0 < ρh (f) < ∞ and g ∼ h. Then lim inf r→∞ log T−1g Tf (r) log T−1 L[h] TL[f] (r) ≤ 1 ≤ lim sup r→∞ log T−1g Tf (r) log T−1 L[h] TL[f] (r) . Proof. From the definition of ρg (f) we get for all sufficiently large values of r that (20) log T−1g Tf (r) ≤ (ρg (f) + ε) log r and for a sequence of values of r tending to infinity that (21) log T−1g Tf (r) ≥ (ρg (f) −ε) log r . Now from (14) and (20) it follows for a sequence of values of r tending to infinity that log T−1g Tf (r) log T−1 L[h] TL[f] (r) ≤ (ρg (f) + ε) log r (ρh (f) −ε) log r . As ε (> 0) is arbitrary we obtain that (22) lim inf r→∞ log T−1g Tf (r) log T−1 L[h] TL[f] (r) ≤ ρg (f) ρh (f) . Now as g ∼ h , in view of Lemma 2.4 and Lemma 2.5 we obtain from (22) that (23) lim inf r→∞ log T−1g Tf (r) log T−1 L[h] TL[f] (r) ≤ 1 . 90 DATTA, BISWAS ABD MONDAL Again combining (5) and (21) we get for a sequence of values of r tending to infinity that log T−1g Tf (r) log T−1 L[h] TL[f] (r) > (ρg (f) −ε) log r (ρh (f) + ε) log r . Since ε (> 0) is arbitrary, it follows that (24) lim sup r→∞ log T−1g Tf (r) log T−1 L[h] TL[f] (r) > ρg (f) ρh (f) . Now as g ∼ h , in view of Lemma 2.4 and Lemma 2.5 we obtain from (24) that (25) lim sup r→∞ log T−1g Tf (r) log T−1 L[h] TL[f] (r) > 1 . Thus the theorem follows from (23) and (25) . � References [1] Bernal, L. : Orden relative de crecimiento de funciones enteras , Collect. Math., Vol. 39 (1988), pp.209-229. [2] Datta, S. K. , Biswas, T. and Ali, S.: Some growth properties of wronskians using their relative order, Journal of Classical Analysis, Vol. 3, No. 1 (2013), pp. 91-99. [3] Hayman, W. K. : Meromorphic Functions, The Clarendon Press, Oxford, 1964. [4] Lahiri, B. K. and Banerjee, D. :Relative order of entire and meromorphic functions, Proc. Nat. Acad. Sci. India, Vol. 69(A) III(1999), pp.339-354. [5] Valiron, G. : Lectures on the General Theory of Integral Functions, Chelsea Publishing Com- pany, 1949. 1Department of Mathematics,University of Kalyani, Kalyani, Dist-Nadia,PIN- 741235, West Bengal, India 2Rajbari, Rabindrapalli, R. N. Tagore Road, P.O.- Krishnagar, Dist-Nadia, PIN- 741101, West Bengal, India 3Dhulauri Rabindra Vidyaniketan (H.S.), Vill +P.O.- Dhulauri , P.S.- Domkal, Dist- Murshidabad , PIN- 742308, West Bengal, India ∗Corresponding author