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Mediterr. J. Math. 10 (2013), 31–39DOI 10.1007/s00009-011-0164-21660-5446/13/010031-9, published online November 30, 2011© 2011 Springer Basel AG

Mediterranean Journalof Mathematics

Approximation by Complex Beta Operatorsof First Kind in Strips of Compact Disks

Sorin G. Gal∗ and Vijay Gupta

Dedicated to the 60th birthday of professor Francesco Altomare

Abstract. In this paper, the exact order of simultaneous approxima-tion and Voronovskaja kind results with quantitative estimate for thecomplex Beta operators of first kind attached to analytic functions instrips of compact disks are obtained. In this way, we put in evidencethe overconvergence phenomenon for this operator, namely the exten-sions of approximation properties with upper and exact quantitativeestimates, from the real interval (0, 1) to strips in compact disks of thecomplex plane of the form SDr(0, 1) = {z ∈ C; |z| ≤ r, 0 < Re(z) < 1}and SDr[a, b] = {z ∈ C; |z| ≤ r, a ≤ Re(z) ≤ b}, with r ≥ 1 and0 < a < b < 1.Mathematics Subject Classification (2010). Primary 30E10 ; Secondary41A25.Keywords. Complex Beta operator of first kind, strip of compact disk,simultaneous approximation, Voronovskaja-type result, exact degrees ofapproximation.

1. Introduction

If f : G → C is an analytic function in the open set G ⊂ C, with D1 ⊂ G(where D1 = {z ∈ C : |z| < 1}), then S. N. Bernstein proved that the com-plex Bernstein polynomials converges uniformly to f in D1 (see e.g., Lorentz[7], p. 88). Exact quantitative estimates and quantitative Voronovskaja-typeresults for these polynomials (see Gal [3]), together with similar results forcomplex Bernstein-Stancu polynomials, complex Kantorovich-Stancu poly-nomials, complex Favard-Szasz-Mirakjan operators, Butzer’s linear combi-nations of complex Bernstein polynomials, complex Baskakov operators andcomplex Balazs-Szabados operators were obtained by the first author in sev-eral recent papers collected by the recent book Gal [5].

∗Corresponding author.

32 S.G. Gal and V. Gupta Mediterr. J. Math.

Furthermore, the approximation properties of the complex Durrmeyer-type operators were studied in Anastassiou-Gal [2], Gal [4] and Mahmudov[9, 10].

The aim of the present article is to obtain approximation results forthe complex Beta operator of first kind, firstly introduced in the case of realvariable in Muhlbach [11] and studied in the real case by e.g. Lupas [8], Khan[6] and Abel-Gupta-Mohapatra [1].

The complex Beta operators will be defined for all n ∈ N and z ∈ C

satisfying 0 < Re(z) < 1, by

Kn(f, z) =1

B(nz, n(1− z))

∫ 1

0

tnz−1(1− t)n(1−z)−1f(t)dt, (1.1)

where B(α, β) is the Euler’s Beta function, defined by

B(α, β) =

∫ 1

0

tα−1(1− t)β−1, α, β ∈ C, Re(α), Re(β) > 0.

Our results put in evidence the overconvergence phenomenon for thiscomplex Beta operator of first kind, that is the extensions of approximationproperties with upper and exact quantitative estimates, from the real interval(0, 1) to strips in compact disks of the complex plane of the form

SDr(0, 1) = {z ∈ C; |z| ≤ r, 0 < Re(z) < 1}and

SDr[a, b] = {z ∈ C; |z| ≤ r, a ≤ Re(z) ≤ b},with r ≥ 1 and 0 < a < b < 1.

2. Auxiliary Results

In the sequel, we shall need the following auxiliary results.

Lemma 2.1. For all ep = tp, p ∈ N ∪ {0}, n ∈ N, z ∈ C with 0 < Re(z) < 1,we have Kn(e0, z) = 1 and

Kn(ep+1, z) =nz + p

n+ pKn(ep, z).

Proof. By the relationship (1.1) of the Beta operators, it is obvious thatKn(e0, z) = 1. Next

Kn(ep+1, z) =1

B(nz, n(1− z))B(nz + p+ 1, n(1− z))

=1

B(nz, n(1− z))

[(nz + p)

Γ(nz + p)Γ(n(1− z))

(n+ p)Γ(n+ p)

]=

nz + p

n+ pKn(ep, z).

This completes the proof of Lemma 2.1. �

Vol. 10 (2013) Complex Beta Type Operators 33

Lemma 2.2. If f is analytic in DR = {z ∈ C; |z| < R}, namely f(z) =∑∞k=0 ckz

k, for all z ∈ DR, then for all n ∈ N, 1 ≤ r < R and z ∈ SDr(0, 1)we have

Kn(f, z) =

∞∑k=0

ck ·Kn(ek, z).

Moreover, for any 0 < a < b < 1, the convergence of the above series isuniform in SDr[a, b].

Proof. Let |z| ≤ r and 0 < Re(z) < 1. Defining fm(z) =∑m

k=0 ckek(z), by thelinearity of Kn given by (1.1), it follows that Kn(fm, z) =

∑mk=0 ckKn(ek, z),

for all |z| ≤ r, 0 < Re(z) < 1 and n,m ∈ N. It suffices to prove that forany fixed n ∈ N, we have limm→∞Kn(fm, z) = Kn(f, z), uniformly in anycompact strip SDr[a, b] with 0 < a < b < 1.

Indeed, we get

|Kn(fm, z)−Kn(f, z)|

=1

|B(nz, n(1− z))| ·∣∣∣∣∫ 1

0

tnz−1(1− t)n(1−z)−1[fm(t)− f(t)]dt

∣∣∣∣≤ 1

|B(nz, n(1− z))| · ‖fm − f‖r∫ 1

0

|tnz−1(1− t)n(1−z)−1|dt.

Because |B(nz, n(1 − z))| > 0 for |z| ≤ r, 0 < Re(z) < 1 and n ∈ N, bythe continuity of |B(nz, n(1 − z))| as function of z, it follows that for any0 < a < b < 1, there exist A,M > 0 both depending on a, b, r, such thatA ≤ |B(nz, n(1 − z))| ≤ M , for all z ∈ SDr[a, b]. This immediately impliesthat there exists a positive constant Ca,b,r,A,M (f) > 0 (independent of m),such that for all z ∈ SDr[a, b] we have

|Kn(fm, z)−Kn(f, z)| ≤ Ca,b,r,A,M (f)‖fm − f‖r, m ∈ N,

which for m → ∞ proves the lemma. �

3. Main Results

The first main result one refers to upper estimate.

Corollary 3.1. Let R > 1 and f(z) =∑∞

k=0 ckzk for all |z| < R. Take

1 ≤ r < R. For all z ∈ SDr(0, 1) and n ∈ N, we have

|Kn(f, z)− f(z)| ≤ Cr(f)

n,

where Cr(f) =1+r2 ·∑∞

k=2 |ck|k(k − 1)rk−1 < ∞.

Proof. Suppose that |z| ≤ r with 0 < Re(z) < 1. By Lemma 2.2 we haveKn(f, z) =

∑∞k=0 ckKn(ek, z). Therefore we get

|Kn(f, z)−f(z)| ≤∞∑k=0

|ck| · |Kn(ek, z)−ek(z)| =∞∑k=2

|ck| · |Kn(ek, z)−ek(z)|,


as Kn(e0, z) = e0(z) = 1 and Kn(e1, z) = e1(z) = z.By using now Lemma 2.1, for all |z| ≤ r, 0 < Re(z) < 1 and n ∈ N, we

get

|Kn(ek+1, z)− ek+1(z)|

=

∣∣∣∣nz + k

n+ kKn(ek)(z)− nz + k

n+ kek(z) +

nz + k

n+ kek(z)− ek+1(z)

∣∣∣∣≤ |nz + k|

n+ k|Kn(ek)(z)− ek(z)|+ |ek(z)| ·

∣∣∣∣nz + k

n+ k− z

∣∣∣∣≤ r |Kn(ek)(z)− ek(z)|+ rk

|k(1− z)|n+ k

≤ r |Kn(ek)(z)− ek(z)|+ rk(1 + r)k

n,

for all k = 0, 1, . . ..Taking above k = 1, 2, . . . , step by step we easily get by recurrence that

|Kn(ek)(z)− ek(z)| ≤ rk−1(1 + r)1

n[1 + 2 + . . .+ (k − 1)]

= rk−1(1 + r)k(k − 1)

2n,

for all |z| ≤ r, 0 < Re(z) < 1 and n ∈ N, which immediately implies thecorollary. �

The following Voronovskaja-type result with a quantitative estimateholds.

Theorem 3.2. Let R > 1 and suppose that f : DR → C is analytic in DR ={z ∈ C : |z| < R} that is we can write f(z) =

∑∞k=0 ckz

k, for all z ∈ DR.For any fixed r ∈ [1, R) and for all |z| ≤ r with 0 < Re(z) < 1 and n ∈ N,we have ∣∣∣∣Kn(f, z)− f(z)− z(1− z)f ′′(z)

2n

∣∣∣∣ ≤ Mr(f)

n2,

where Mr(f) =∑∞

k=2 |ck|(1 + r)k(k + 1)(k − 1)2rk−1 < ∞.

Proof. Denoting πk,n(z) = Kn(ek)(z) and

Ek,n(z) = πk,n(z)− ek(z)− zk−1(1− z)k(k − 1)

2n,

firstly it is clear that E0,n(z) = E1,n(z) = 0. Then, we can write∣∣∣∣Kn(f, z)− f(z)− z(1− z)f ′′(z)2n

∣∣∣∣ ≤∞∑k=2

|ck| · |Ek,n(z)|,

so it remains to estimate Ek,n(z) for k ≥ 2.In this sense, simple calculation based on Lemma 2.1 too, leads us to

the formula

Ek,n(z) =nz + k − 1

n+ k − 1· Ek−1,n(z) +

zk−2(1− z)(k − 1)2

2n(n+ k − 1)· [k(1− z)− 2].


This immediately implies, for all k ≥ 2 and |z| ≤ r with 0 < Re(z) < 1

|Ek,n(z)| ≤ r|Ek−1,n(z)|+ rk−2(1 + r)(k − 1)2

2n2· [k(1 + r) + 2]

≤ r|Ek−1,n(z)|+ rk−1(1 + r)(k − 1)2(k + 1)

n2.

Taking in the last inequality, k = 2, 3, . . . , and reasoning by recurrence, finallywe easily obtain

|Ek,n(z)| ≤ rk−1(1 + r)

n2

⎡⎣ k∑j=1

(j − 1)2(j + 1)

⎤⎦ ≤ rk−1(1 + r)

n2·k(k−1)2(k+1).

We conclude that∣∣∣∣Kn(f, z)− f(z)− z(1− z)f ′′(z)2n

∣∣∣∣ ≤∞∑k=2

|ck| · |Ek,n|

≤ 1

n2

∞∑k=2

|ck|(1 + r)k(k − 1)2(k + 1)rk−1.

As f (4)(z) =∑∞

k=4 ckk(k − 1)(k − 2)(k − 3)zk−4 and the series is absolutelyconvergent in |z| ≤ r, it easily follows that

∑∞k=4 |ck|k(k − 1)(k − 2)(k −

3)rk−4 < ∞, which implies that∑∞

k=2 |ck|(1 + r)k(k − 1)2(k + 1)rk−1 < ∞.This completes the proof of the theorem. �

In what follows, we obtain the exact order in approximation by thistype of complex Beta operators of first kind and by their derivatives. In thissense, we present the following three results.

Theorem 3.3. Let R > 1 and suppose that f : DR → C is analytic in DR, thatis we can write f(z) =

∑∞k=0 ckz

k, for all z ∈ DR. If f is not a polynomialof degree ≤ 1, then for any r ∈ [1, R) and any 0 < a < b < 1, we have

||Kn(f, ·)− f ||SDr[a,b] ≥ Cr,a,b(f)

n, n ∈ N,

where SDr[a, b] = {z ∈ C : |z| ≤ r, a ≤ Re(z) ≤ b}, ‖f‖SDr[a,b] =sup{|f(z)|; z ∈ SDr[a, b]} and Cr,a,b(f) depends only on f , a, b and r.

Proof. For all |z| ≤ r with 0 < Re(z) < 1 and n ∈ N, we have

Kn(f, z)− f(z)

=1

n

[z(1− z)f ′′(z)

2+

1

n

{n2

(Kn(f, z)− f(z)− z(1− z)f ′′(z)

2n

)}].

Also, we have

||F +G||SDr[a,b] ≥∣∣||F ||SDr[a,b] − ||G||SDr[a,b]

∣∣ ≥ ||F ||SDr[a,b] − ||G||SDr[a,b].


It follows

||Kn(f, ·)− f ||SDr[a,b] ≥ 1

n

[∥∥∥∥e1(1− e1)f′′

2

∥∥∥∥SDr[a,b]

− 1

n

{n2

∣∣∣∣∣∣∣∣Kn(f, ·)− f − e1(1− e1)f

′′

2n

∣∣∣∣∣∣∣∣SDr[a,b]

}].

Taking into account that by hypothesis f is not a polynomial of degree ≤ 1in DR, we get ||e1(1− e1)f

′′||SDr[a,b] > 0.

Indeed, supposing the contrary it follows that z(1− z)f ′′(z) = 0 for allSDr[a, b]. Therefore we get f ′′(z) = 0, for all z ∈ SDr[a, b]. Because f isanalytic in DR, by the uniqueness of analytic functions we get f ′′(z) = 0,for all z ∈ DR, that is f is a linear function in DR, which contradicts thehypothesis.

Now by Theorem 3.2, we have

n2

∣∣∣∣∣∣∣∣Kn(f, ·)− f − e1(1− e1)f

′′

2n

∣∣∣∣∣∣∣∣SDr[a,b]

≤ Mr(f).

Therefore there exists an index n0 depending only on f , a, b and r, such thatfor all n ≥ n0, we have

∥∥∥∥e1(1− e1)f′′

2

∥∥∥∥SDr[a,b]

− 1

n

{n2

∣∣∣∣∣∣∣∣Kn(f, ·)− f − e1(1− e1)f

′′

2n

∣∣∣∣∣∣∣∣SDr[a,b]

}

≥ 1

4‖e1(1− e1)f

′′‖SDr[a,b],

which immediately implies

||Kn(f, ·)− f ||SDr[a,b] ≥ 1

4n||e1(1− e1)f

′′||SDr[a,b], ∀n ≥ n0.

For n ∈ {1, 2, . . . , n0−1} we obviously have ||Kn(f, ·)−f ||SDr[a,b] ≥ Mr,a,b,n(f)n

with Mr,a,b,n(f) = n||Kn(f, ·) − f ||SDr[a,b] > 0. Indeed, if we would have||Kn(f, ·) − f ||SDr[a,b] = 0, then would follow Kn(f, z) = f(z) for all z ∈SDr[a, b], which is valid only for f a linear function, contradicting the hy-pothesis on f .

Therefore, finally we obtain ||Kn(f, ·) − f ||SDr[a,b] ≥ Cr,a,b(f)n for all n,

where

Cr,a,b(f) =

= min{Mr,a,b,1(f),Mr,a,b,2(f), . . . ,Mr,a,b,n0−1(f),1

4||e1(1− e1)f

′′||SDr[a,b]},which completes the proof. �

As a consequence of Corollary 3.1 and Theorem 3.3, we have the follow-ing:


Corollary 3.4. Let R > 1 and suppose that f : DR → C is analytic in DR.If f is not a polynomial of degree ≤ 1, then for any r ∈ [1, R) and any0 < a < b < 1, we have

||Kn(f, ·)− f ||SDr[a,b] ∼ 1

n, n ∈ N,

where the constants in the equivalence depend only on f , a, b and r.

Our last result is in simultaneous approximation and can be stated asfollows.

Theorem 3.5. Let R > 1 and suppose that f : DR → C is analytic in DR i.e.f(z) =

∑∞k=0 ckz

k, for all z ∈ DR, 1 ≤ r < r1 < R, 0 < a1 < a < b < b1 < 1and p ∈ N be fixed. If f is not a polynomial of degree ≤ max{1, p− 1}, thenwe have

||K(p)n (f, ·)− f (p)||SDr[a,b] ∼ 1

n, n ∈ N,

where the constants in the equivalence depend only on f, r, r1, a, b, a1, b1 andp.

Proof. Denote by Γ = Γa1,b1,r1 = S1

⋃A1

⋃S2

⋃A2 the closed curve com-

posed by the segments in C

S1 =

{z = x+ iy ∈ C;x = a1 and −

√r21 − a21 ≤ y ≤

√r21 − a21

},

S2 =

{z = x+ iy ∈ C;x = b1 and −

√r21 − b21 ≤ y ≤

√r21 − b21

},

and by the arcs A1, A2 on the circle of center origin and radius r1, situatedin the region between the two segments defined above.

Clearly that Γ together with its interior is exactly SDr1 [a1, b1] and thatfrom r < r1 we have SDr[a, b] ⊂ SDr1 [a1, b1], the inclusion beng strictly.

By the Cauchy’s integral formula for derivatives, we have for all z ∈SDr[a, b] and n ∈ N

K(p)n (f, z)− f (p)(z) =

p!

2πi

∫Γ

Kn(f, u)− f(u)

(u− z)p+1du,

which by Corollary 3.1 and by the inequality |u − z| ≥ d = min{r1 − r, a −a1, b1 − b} valid for all z ∈ SDr[a, b] and u ∈ Γ, implies

||K(p)n (f, ·)− f (p)||SDr[a,b] ≤ p!

2π.l(Γ)

dp+1||Kn(f, ·)− f ||SDr1 [a,b]

≤ 1

n· Cr1(f)p!l(Γ)

2πdp+1.

Note that here, by simple geometrical reasonings, for the length l(Γ) of Γ, weget

l(Γ) = l(S1) + l(S2) + l(A1) + l(A2)

= 2(√r21 − a21 +

√r21 − b21) + 2r1[arccos(a1/r1)− arccos(b1/r1)],

where arccos(α) is considered expressed in radians.


It remains to prove the lower estimation for ||K(p)n (f, ·)− f (p)||SDr[a,b].

By the proof of Theorem 3.3, for all u ∈ Γ and n ∈ N, we have

Kn(f, u)− f(u) =1

n

[u(1− u)f ′′(u)

2

+1

n

{n2

(Kn(f, u)− f(u)− u(1− u)f ′′(u)

2n

)}].

Substituting it in the above Cauchy’s integral formula, we get

K(p)n (f, z)− f (p)(z) =

1

n

[(z(1− z)f ′′(z)

2

)(p)

+1

n· p!

2πi

∫Γ

n2(Kn(f, u)− f(u)− u(1−u)f ′′(u)

2n

)(u− z)p+1

du

].

Thus

∣∣∣∣∣∣K(p)n (f, ·)− f (p)

∣∣∣∣∣∣SDr[a,b]

≥ 1

n

[∥∥∥∥∥[e1(1− e1)f

′′

2

](p)∥∥∥∥∥SDr[a,b]

− 1

n

∣∣∣∣∣∣∣∣∣∣∣∣p!

2πi

∫Γ

n2(Kn(f, u)− f(u)− u(1−u)f ′′(u)

2n

)(u− ·)p+1

du

∣∣∣∣∣∣∣∣∣∣∣∣SDr[a,b]

].

Applying Theorem 3.2 too, it follows∣∣∣∣∣∣∣∣∣∣∣∣p!

2πi

∫Γ

n2(Kn(f, u)− f(u)− u(1−u)f ′′(u)

2n

)(u− ·)p+1

du

∣∣∣∣∣∣∣∣∣∣∣∣SDr[a,b]

≤ p!

2π

l(Γ)n2

dp+1

∣∣∣∣∣∣∣∣Kn(f, ·)− f − e1(1− e1)f

′′

2n

∣∣∣∣∣∣∣∣SDr1 [a1,b1]

≤ Mr1(f)l(Γ)p!

2πdp+1.

But by the hypothesis on f , we necessarily have

||[e1(1− e1)f′′/2](p)||SDr[a,b] > 0.

Indeed, supposing the contrary we get that e1(1−e1)f′′ is a polynomial

of degree ≤ p− 1 in SDr[a, b], which by the uniqueness of analytic functionsimplies that

z(1− z)f ′′(z) = Qp−1(z) for all z ∈ DR,

where Qp−1(z) is a polynomial of degree ≤ p− 1.Now, if p = 1 and p = 2, then the analyticity of f in DR easily implies

that f necessarily is a polynomial of degree ≤ 1 = max{1, p−1}. If p > 2, thenthe analyticity of f in DR easily implies that f necessarily is a polynomialof degree ≤ p − 1 = max{1, p − 1}. Therefore, in all the cases we get acontradiction with the hypothesis.

In conclusion, ||[e1(1−e1)f′′/2](p)||SDr[a,b] > 0 and in continuation, rea-

soning exactly as in the proof of Theorem 3.3, but for ‖K(p)n (f, ·)−f (p)‖SDr[a,b]


instead of ‖Kn(f, ·) − f‖SDr[a,b], we immediately get the desired conclu-sion. �Acknowledgment

The authors thank the referee for the very useful remarks.

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[6] M. K. Khan, Approximation properties of Beta operators, in : Progress in Ap-proximation Theory, Academic Press, New York, 1991, pp. 483-495.

[7] G. G. Lorentz, Bernstein Polynomials, Chelsea Publ., Second edition, NewYork 1986.

[8] A. Lupas, Die Folge der Beta-Operatoren, Dissertation, Univ. Stuttgart,Stuttgart, 1972.

[9] N. I. Mahmudov, Approximation by genuine q-Bernstein-Durrmeyer polynomi-als in compact disks, Hacet. J. Math. Stat. 40(2011), No. 1 77-89.

[10] N. I. Mahmudov, Approximation by Bernstein–Durrmeyer-type operators incompact disks, Appl. Math. Lett. 24(2011), No. 7, 1231-1238.

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Sorin G. GalDepartment of Mathematics and Computer Science, University of OradeaStr. Universtatii No. 1, 410087 Oradea, Romaniae-mail: [email protected]

Vijay GuptaSchool of Applied Sciences, Netaji Subhas Institute of TechnologySector 3 Dwarka, New Delhi-110078, Indiae-mail: [email protected]

Received: April 4, 2011.Revised: July 22, 2011.Accepted: September 6, 2011.