The impact of the limit $q$-Durrmeyer operator on continuous functions

The limit $q$-Durrmeyer operator, $D_{\infty,q},$ was introduced and its approximation properties were investigated by V. Gupta in 2008 during a study of $q$-analogues for the Bernstein-Durrmeyer operator. In the present work, this operator is investigated from a different perspective. More precisely, the growth estimates are derived for the entire functions comprising the range of $D_{\infty,q}$. The interrelation between the analytic properties of a function $f$ and the rate of growth for $D_{\infty,q}f$ are established, and the sharpness of the obtained results are demonstrated.


Introduction
The significant influence of the Bernstein polynomials on modern mathematics -not only theoretical, but also applied and computational -brought about the emergence of its numerous versions and modifications.See, for example, [2,3,12].While the Bernstein polynomials serve to approximate the continuous functions on [0, 1], the Kantorovich polynomials constructed with respect to the Bernstein basis are applicable for the approximation of integrable functions.Kantorovich's breakthrough idea was further developed by Durrmeyer [7] and Derriennic [6].The latter proved that the Bernstein-Durrmeyer polynomials approximate functions in L 1 [0, 1], and also generate self-adjoint operators in L 2 [0, 1].
With the increasing role of the q-Calculus (see, e.g.[1,4,5,15]), the q-analogues of various Bernstein-type operators have come to the fore.The reader is referred to [3,8,14].New versions of these operators, targeting a wide spectrum of various problems, are continuously coming out.
In 2008, V. Gupta [9] introduced a simple q-analogue of the Bernstein-Durrmeyer operators, denoted by D n,q , and studied its approximation properties.One of the properties that he proved was that {D n,q } converges to the limit operator D ∞,q in the strong operator topology on C[0, 1].More results on the q-Durrmeyer operator have been obtained in [10,13].
In the present work, further investigation is carried out concerning the limit q-Bernstein-Durrmeyer operator.Distinct from the preceding studies on the subject, this paper is focused on the analytic properties that the image of f ∈ C[0, 1] possesses under the operator D ∞,q .Here, it is proved that, for each f ∈ C[0, 1], the function D ∞,q f admits an analytic continuation from [0, 1] to the whole complex plane C. The growth estimates of the entire function D ∞,q f are provided, along with the interconnection between the growth of D ∞,q f and the behaviour of f .The sharpness of the obtained results is demonstrated.
To present the results, let us recollect the necessary notation and definitions.The q-Pochhammer symbol denotes, for each a ∈ C, The Euler Identities will be used permanently.See [1, Chapter 10, Corollary 10.2.2].The q-integral over an interval [0, a], first introduced by Thomae [16] and later by Jackson [11], is defined as (1.3) Definition 1.1.[9] Let q ∈ (0, 1), f ∈ C[0, 1].The limit q-Durrmeyer operator is defined by where and As coefficients (1.4) form a bounded sequence whatever f ∈ C[0, 1] is, the function D ∞,q f admits an analytic continuation from [0, 1] to open disc {z : |z| < 1}.Taking into account (1.3),A ∞k (f ) can also be expressed as f (q j )q (k+1)j (q; q) j . (1.6) Throughout the paper, letter C -with or without subscripts -denotes a positive constant whose specific value is of no importance.Subscripts, when used, indicate the dependence of C on certain parameters.It should be pointed out that the same letter may stand for different values.Moreover, if f is analytic in the closed disc ∆ r := {z : |z| r}, the notation The article is organized as follows: In Section 2, the main outcomes are stated, while Section 3 contains the auxiliary technical lemmas.Finally, the proofs of the main results appear in Section 4.

Statement of Results
Theorem 2.1.For each f ∈ C[0, 1], the function (D ∞,q f )(x) admits an analytic continuation from [0, 1] as an entire function given by (q; q) n (z; q) n+j . (2.1) The proof of Theorem 2.1 presented in Section 4 yields, apart from (2.1), the following corollary: Corollary 2.2.The growth of D ∞,q f , for each f ∈ C[0, 1], enjoys the following estimate: It is worth pointing out that coefficients (1.6) can be viewed as the values of the function g(z) := (qz; q) ∞ ρ(z) at points z = q k , k = 0, 1, . .., where f (q j )q j (q; q) j z j .
(2.3) Since (qz; q) ∞ is entire and the series converges in the disc {z : |z| < 1/q} for any f ∈ C[0, 1], it follows that g is analytic in that disc.Clearly, the radius of convergence for ρ can be greater than 1/q.The representation below of D ∞,q with the help of divided differences of g is important.
This representation allows us to not only refine the estimate of Corollary 2.2, but also establish a connection between the behaviour of f and the growth of its image under D ∞,q .
As a consequence of Theorem 2.4, the crude estimate (2.2) can be improved.Since ρ is analytic in {z : |z| < 1/q}, it is possible to assume λ = 0 in Theorem 2.4 and obtain the following result. ) Indeed, in this case, ρ is analytic in {z : |z| < q −1−α }.
The estimate in Theorem 2.4 is sharp as demonstrated by the assertion below.
Theorem 2.8.For every λ > 1, there exists f ∈ C[0, 1] such that Theorem 2.4 and Corollaries 2.5-2.7 establish the connection between the radius of convergence for series (2.3) and the rate of growth for D ∞,q f.In the general sense, the greater the radius is, the slower the growth becomes.Approaching the problem from a different angle, the dependence of the growth on the differentiability of f at the origin is addressed in the next assertion.The statement makes it possible to obtain better estimates for M(r; D ∞,q f ) than those guaranteed by Theorem 2.4 when f is differentiable at 0 even though the series (2.3) converges solely in the smallest admissible disc.Theorem 2.9.Let f be m times differentiable at 0 from the right.Then, for all λ < 1 + m.
Corollary 2.10.If f is infinitely differentiable at 0 from the right, then (2.5) holds for all λ > 0. In particular, (2.5) is valid whenever f is analytic in a neighbourhood of 0.

Auxiliary Results
In what comes next, the function τ given by Proof.
By virtue of (1.2), it follows that Consequently, one obtains Hence, τ (z) is analytic in ∆ R for each R > 0 and (3.1) is valid for all z ∈ C. Therefore, τ (z) is an entire function.
Proof.It is known that (see for example, [12, Section 2.7., p.44, Eq. ( 4)]) where L is a positively-oriented, simple and closed curve encircling the distinct points a 0 , . . ., a k and g is analytic anywhere on and inside L.
Proof of Theorem 2.9.By Taylor's Theorem, one can write where T m (x) is a polynomial of degree at most m and S m (x) = o(x m ) as x → 0 + .Since D ∞,q maps a polynomial to a polynomial of the same degree (see [9,Remark 3]), there holds (D ∞,q f )(z) = P m (z) + (D ∞,q S m )(z), where P m (z) is a polynomial of degree at most m and, as such, M(r; P m ) = o(r −λ (−r; q) ∞ ), r → ∞, for all λ > 0. As for M(r; D ∞,q S m ), it can be estimated by means of Corollary 2.6 with α = m.

plays a key role. Lemma 3 . 1 .
The function τ admits an analytic continuation from the open unit disc as an entire function.