BOUNDS FOR FUNCTIONALS DEFINED ON A CERTAIN CLASS OF MEROMORPHIC FUNCTIONS

Abstract We obtain bounds for certain functionals defined on a class of meromorphic functions in the unit disc of the complex plane with a nonzero simple pole. These bounds are sharp in a certain sense. We also discuss possible applications of this result. Finally, we generalise the result to meromorphic functions with more than one simple pole.


Introduction and main result
We denote the set of all complex numbers by C. Let A be the class of analytic functions in the unit disc D := {z ∈ C : |z| < 1} with the Taylor expansion and f (z) 0 for z ∈ D \ {0}.We see that the functions in A must satisfy the normalisation f (0) = 0 = f (0) − 1.For f ∈ A, we define and for f ∈ A that are bounded in D, let Lewin obtained the following result.
THEOREM 1.1 (Lewin,[4]).For f ∈ A with the expansion (1.1) 2 B. Bhowmik and F. Parveen [2] In the same article, Lewin established that the estimates in Theorem 1.1 are best possible.He commented that although the bounds in Theorem 1.1 are not sharp in the case of univalent or bounded univalent functions, they nevertheless supply information which may be of help when dealing with conformal mappings (analytic and univalent mappings).
In this article, we allow the functions in A to possess a nonzero simple pole inside D and wish to see whether an analogue of Theorem 1.1 can be established after suitably defining the quantities d f and D f in this case.Therefore, we consider functions f that are meromorphic having a simple pole at z = p ∈ (0, 1) inside the unit disk D, with the Taylor series expansion where D p := {z ∈ C : |z| < p} and such that f does not vanish in D other than at the origin.Evidently, for such f, we have f (0 We denote the class of such functions by F (p).If g is a meromorphic function having a simple pole at pe iβ , β ∈ (0, 2π], p ∈ (0, 1), and g is nonvanishing in D \ {0} with g(0) = 0 and g (0) 0, then This shows that taking the pole p in the interval (0, 1) is sufficiently general.For f ∈ F (p), we define and if (z − p) f is bounded in D, we define These quantities can be thought of as analogous to d f and D f in [4].The reason for multiplying f /z, f ∈ F (p), by the factor (z − p)(1 − pz) is to make the resulting function holomorphic in D. In addition, if f has a holomorphic extension to the boundary Thus, in such cases, finding the bounds of d p ( f ) and D p ( f ) will essentially give estimates for the distance between the origin and the image of the unit circle under f.In the second part of this paper, we will generalise these results to functions having more than one simple pole in D.
We now state and prove our main result.We will adopt the main idea of the proof from [4], but as we approach the problem, we will realise that the proof itself and [3] Bounds for functionals 3 finding the extremal functions for which equalities hold in these estimates are not straightforward.
and if These bounds are best possible.
Then we must have where we choose that branch of logarithm for which log f (0) = 0.A minor simplification of the above inequality yields

Now define
which is analytic in D by choosing that branch of the logarithm for which log( f (0)) = 0.By virtue of the previous inequality, we have Re F(z) ≥ 0 with F(0) = 1.Now we can expand F about the origin to get An application of Caratheodory's lemma (see [3]) for the function F in D p yields Letting d p ( f ) = p/s gives the first estimate of the theorem.To obtain the second estimate of the theorem, we let Note that g ∈ F (p) as (z − p) f is bounded in D and g has the Taylor expansion B. Bhowmik and F. Parveen [4] We thus have d p (g)/p = p/D p ( f ).Therefore, Consequently, the second inequality of the theorem follows.
The bounds obtained in the theorem are best possible in the following sense.We consider the functions f ± α in F (p): where Therefore, here a 2 = α/p + 1/p − p, which gives |a 2 p + p 2 − 1| = α and where φ ∈ (0, π) ∪ (π, 2π).Again for the function f − α , we have where φ ∈ (0, π) ∪ (π, 2π).This shows that the estimates stated in the theorem are best possible.This completes the proof of the theorem.
REMARK 1.3.(i) Note that the quantity |pa 2 + p 2 − 1| in the bounds for d p ( f ) and D p ( f ) in Theorem 1.2, may be replaced by |p 2 (a 3 − 1 2 a 2 2 ) + (p 4 − 1)/2| as by Caratheodory's lemma, we also have for the function F defined in D p .Furthermore, we comment here that if pa 2 + p 2 − 1 = 0, then we need to use the first nonvanishing coefficient in the expansion (1.4) to get the estimates for d p ( f ) and D p ( f ).
(ii) We observe that we recover Lewin's results (compare [4, Theorem A]) if we pass to the limit as p → 1− in the expression for the bounds obtained in Theorem 1.2.
We now illustrate the results obtained in Theorem 1.2 through some examples and indicate possible applications of the bounds.
We choose the branch of the logarithm such that log 1 = 0.One can check that f ∈ F (p) and has the expansion Here, a 2 = 1/p − 1/2 and as a result, an application of Theorem 1.2 yields It is a simple exercise to check that f is one-one in D (see [1,2]).The Taylor expansion of this function yields the second Taylor coefficient as a 2 = p + 1/p.Therefore, according to Theorem 1.2, we must have d p ( f ) ≤ p exp(−p 2 ).Now for this function, for all p ∈ (0, 1).Thus, the obtained bound in Theorem 1.2 is not sharp for this univalent function.
In the above three examples, it is difficult to give the exact estimate for the distance from the origin to the image of the unit circle under f, but nonetheless, we obtain some information about this distance.EXAMPLE 1.7 (Existence of a zero).As an application of Theorem 1.2, we wish to investigate the existence of a zero for a meromorphic function f with a nonzero pole other than at the origin.To this end, consider p = 1/2 and the function Suppose f /z does not vanish in D \ {0}.Then it is clear that f ∈ F (1/2).Expanding f in a Taylor series about the origin for |z| < 1/2 gives Here, a 2 = 17.Therefore, an application of Theorem 1.2 yields However, then we see that This is a contradiction, and therefore f /z must vanish at a nonzero point in D.

Generalisation of the main result
In this section, we generalise Theorem 1.2 by allowing the functions in F (p) to have more than one nonzero simple pole in D. This extension is possible if these poles in D lie on a line passing through the origin, that is, all the poles have the same argument.Thus, it will be sufficient to consider these nonzero poles in the interval (0, 1) as we did for one nonzero pole in D (see (1.3)).More precisely, we consider functions f that are meromorphic having simple poles at z = p 1 , p 2 , . . ., p n ∈ (0, 1) inside the unit disk D with the Taylor series expansion In the next theorem, we obtain estimates for m p ( f ) and M p ( f ).These bounds are best possible.