Computers and Mathematics with Applications 56 (2008) 1121–1127www.elsevier.com/locate/camwa

Approximation and geometric properties of complexFavard–Szasz–Mirakjan operators in compact disks

Sorin G. Gal

Department of Mathematics and Computer Science, University of Oradea, Street Universitatii No. 1, 410087 Oradea, Romania

Received 26 June 2007; received in revised form 22 January 2008; accepted 6 February 2008

Abstract

In this paper, first we obtain quantitative estimates of the convergence and of the Voronovskaja’s theorem in compact disks,for complex Favard–Szasz–Mirakjan operators attached to analytic functions satisfying some suitable exponential-type growthcondition. Then, we prove that beginning with an index, these operators preserve the starlikeness, convexity and spirallikeness inthe unit disk.c© 2008 Elsevier Ltd. All rights reserved.

Keywords: Complex Favard–Szasz–Mirakjan operators; Quantitative estimates; Voronovskaja’s theorem; Starlikeness; Convexity and spirallikeness

1. Introduction

In addition to the convergence results in Lorentz [1], in [2] and [3] we have obtained estimates for the convergenceand Voronovskaja’s theorem, of complex Bernstein polynomials defined by Bn( f )(z) =

∑nk=0

( nk

)zk(1 −

z)n−k f (k/n) and attached to an analytic function f in closed disks. Also, in [2] we proved shape-preserving propertiesof complex Bernstein polynomials.

The goal of the present note is to extend these type of results to complex Favard–Szasz–Mirakjan operators.Section 2 deals with approximation properties of these operators, in Section 3 we obtain a Voronovskaja result with

quantitative estimate, while in Section 4 one prove some shape-preserving properties for these operators.

2. Approximation results

For a real function of real variable f : [0, ∞) → R, it is well known that the Favard–Szasz–Mirakjan operators

are given by Sn( f )(x) = e−nx ∑∞

j=0(nx) j

j ! f ( j/n), x ∈ [0, ∞), where for the convergence of Sn( f )(x) to f (x),

usually f is supposed to be of exponential growth, that is | f (x)| ≤ CeBx , for all x ∈ [0, +∞), with C, B > 0 (seeFavard [4]). Also, concerning quantitative estimates in approximation of f (x) by Sn( f )(x), in e.g. [5], it is provedthat under some additional assumptions on f , we actually have |Sn( f )(x) − f (x)| ≤

Cn , for all x ∈ R+, n ∈ N.

E-mail address: galso@uoradea.ro.

0898-1221/$ - see front matter c© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.camwa.2008.02.014

1122 S.G. Gal / Computers and Mathematics with Applications 56 (2008) 1121–1127

The complex Favard–Szasz–Mirakjan operator is obtained from the real version, simply replacing the real variablex by the complex one z, that is

Sn( f )(z) = e−nz∞∑j=0

(nz) j

j !f ( j/n).

In this section, supposing that f : [0, +∞) → C of exponential growth, can be prolonged to an analytic functionin an open disk (with center in origin) by keeping exponential growth, we obtain quantitative estimates in closed diskswith center in origin, similar in form with that in the real case in [5] mentioned above.

Let us note that the first result concerning the convergence of complex Sn( f )(z) to f (z) belonging to a class ofanalytic functions satisfying a suitable exponential-type growth condition in a parabolic domain, was proved in [6],but without any estimate of the approximation error.

The main result of this section can be summarized by the following

Theorem 2.1. Let DR = {z ∈ C; |z| < R} be with 1 < R < +∞ and suppose that f : [R, +∞) ∪ DR → C iscontinuous in [R, +∞)∪DR , analytic in DR , i.e. f (z) =

∑∞

k=0 ck zk , for all z ∈ DR , and that there exist M, C, B > 0

and A ∈ ( 1R , 1), with the property |ck | ≤ M Ak

k!, for all k = 0, 1, . . ., (which implies | f (z)| ≤ MeA|z| for all z ∈ DR)

and | f (x)| ≤ CeBx , for all x ∈ [R, +∞).(i) Let 1 ≤ r < 1

A be arbitrary fixed. For all |z| ≤ r and n ∈ N, we have

|Sn( f )(z) − f (z)| ≤Cr,A

n,

where Cr,A =M2r

∑∞

k=2(k + 1)(r A)k < ∞.(ii) For the simultaneous approximation by complex Favard–Szasz–Mirakjan operators, we have : if 1 ≤ r < r1 < 1

Aare arbitrary fixed, then for all |z| ≤ r and n, p ∈ N,

|S(p)n ( f )(z) − f (p)(z)| ≤

p!r1Cr1,A

n(r1 − r)p+1 ,

where Cr1,A is given as at the above point (i).

Proof. (i) According to Theorem 2 in [7], we can write

Sn( f )(z) =

∞∑j=0

[0, 1/n, . . . , j/n; f ]z j ,

where [0, 1/n, . . . , j/n; f ] denotes the divided difference of f on the knots 0, 1/n, . . . , j/n. Note that the aboveformula was proved in [7] for functions of real variable, but the formula holds in complex setting too, since onlyalgebraic calculations were used (see the proof of Theorem 2 in [7]).

Taking in this representation formula ek(z) = zk , we obtain that Tn,k(z) := Sn(ek)(z) is a polynomial of degree≤ k, k = 0, 1, 2, . . ., and Tn,0(z) = 1, Tn,1(z) = z, for all z ∈ C. Also, differentiating Tn,k(z) with respect to z 6= 0,we get

T ′

n,k(z) =

∞∑j=0

jk

nk

[−ne−nz (nz) j

j !+ e−nz jn

(nz) j−1

j !

]

= −nTn,k(z) +

∞∑j=0

jk+1

nk+1 e−nz n

z

(nz) j

j != −nTn,k(z) +

n

zTn,k+1(z),

which implies

Tn,k+1(z) =z

nT ′

n,k(z) + zTn,k(z),

for all z ∈ C, k ∈ {0, 1, 2, . . . , }, n ∈ N. From this it is immediate the recurrence formula

Tn,k(z) − zk=

z

n[Tn,k−1(z) − zk−1

]′+ z[Tn,k−1(z) − zk−1

] +k − 1

nzk−1,

for all z ∈ C, k, n ∈ N.

S.G. Gal / Computers and Mathematics with Applications 56 (2008) 1121–1127 1123

Now, let 1 ≤ r < R. Denoting with ‖ · ‖r the norm in C(Dr ), where Dr = {z ∈ C; |z| ≤ r}, by a lineartransformation, Bernstein’s inequality in the closed unit disk becomes |P ′

k(z)| ≤kr ‖Pk‖r , for all |z| ≤ r , where Pk(z)

is a polynomial of degree ≤ k. Therefore, from the above recurrence formula, we get

‖Tn,k − ek‖r ≤r

n· ‖Tn,k−1 − ek−1‖r

k − 1r

+ r‖Tn,k−1 − ek−1‖r +rk−1(k − 1)

n,

which implies the recurrence

‖Tn,k − ek‖r ≤

(r +

k − 1n

)‖Tn,k−1 − ek−1‖r +

rk−1(k − 1)

n.

In what follows we prove by mathematical induction with respect to k (with n ≥ 1 supposed to be fixed, arbitrary),that this recurrence implies

‖Tn,k − ek‖r ≤(k + 1)!

2nrk−1, for all k ≥ 2, n ≥ 1.

Indeed, for k = 2 and n ∈ N, the left-hand side is rn while the right-hand side is 3r

n . Supposing now that it is truefor k, the above recurrence implies

‖Tn,k+1 − ek+1‖r ≤

(r +

k

n

)(k + 1)!

2nrk−1

+rkk

n.

It remains to prove that(r +

k

n

)(k + 1)!

2nrk−1

+rkk

n≤

(k + 2)!

2nrk,

or, after simplifications, equivalently to(r +

k

n

)(k + 1)! + 2rk ≤ (k + 2)!r.

It is easy to see that this last inequality holds true for all k ≥ 2 and n ∈ N.Now, from the hypothesis on f (that is | f (x)| ≤ max{M, C}emax{A,B}x , for all x ∈ R+), it follows that (see e.g. [6],

pp. 1171-1172 and p. 1178) Sn( f )(z) is analytic in DR . Therefore, it is easy to see that we can write

Sn( f )(z) =

∞∑k=0

ck Sn(ek)(z) =

∞∑k=0

ck Tn,k(z), for all z ∈ DR,

which from the hypothesis on ck , immediately implies for all |z| ≤ r

|Sn( f )(z) − f (z)| ≤

∞∑k=2

|ck | · |Tk,n(z) − ek(z)| ≤

∞∑k=2

MAk

k!

(k + 1)!

2nrk−1

=M

2nr

∞∑k=2

(k + 1)(r A)k=

Cr,A

n,

where Cr,A =M2r

∑∞

k=2(k + 1)(r A)k < ∞, for all 1 ≤ r < 1A , taking into account that the series

∑∞

k=2 uk+1 andtherefore its derivative

∑∞

k=2(k + 1)uk , are uniformly and absolutely convergent in any compact disk included in theopen unit disk.

(ii) Denoting by γ the circle of radius r1 > r and center 0, since for any |z| ≤ r and v ∈ γ , we have |v−z| ≥ r1−r ,by Cauchy’s formulas it follows that for all |z| ≤ r and n ∈ N, we have

|S(p)n ( f )(z) − f (p)(z)| =

p!

2π

∣∣∣∣∫γ

Sn( f )(v) − f (v)

(v − z)p+1 dv

∣∣∣∣≤

Cr1,A

n

p!

2π

2πr1

(r1 − r)p+1 =Cr1,A

n

p!r1

(r1 − r)p+1 ,

which proves (ii) and the theorem. �

1124 S.G. Gal / Computers and Mathematics with Applications 56 (2008) 1121–1127

Corollary 2.2. In the hypothesis of Theorem 2.1, if f is not a polynomial of degree ≤ 1 in the case (i) and if f is nota polynomial of degree ≤ p in the case (ii) , then 1

n is in fact the exact order of approximation.

Proof. Applying the norm ‖ · ‖r to the identity

Sn( f )(z) − f (z) =1n

{z

2f ′′(z) +

1n

[n2

(Sn( f )(z) − f (z) −

z

2nf ′′(z)

)]},

it follows

‖Sn( f ) − f ‖r ≥1n

{∥∥∥e1

2f ′′

∥∥∥r−

1n

[n2

∥∥∥Sn( f ) − f −e1

2nf ′′

∥∥∥r

]}.

If f is not a polynomial of degree ≤ 1 then evidently ‖e12 f ′′

‖r > 0, which combined with the estimate inTheorem 3.2 immediately implies that ‖Sn( f ) − f ‖r ≥

Cn , for all n ≥ n0, with C > 0 independent of n. Since

for n = 1, 2, . . . , n0 − 1 the inequality ‖Sn( f ) − f ‖r ≥C1n is trivial with a constant C1 > 0 and taking into account

the upper estimate in Theorem 2.1, (i), we obtain the desired conclusion.Now, replacing Sn( f )(z) − f (z) in the above identity, to the Cauchy formula in the proof of Theorem 2.1, (ii) and

then applying the norm ‖ · ‖r to the integral identity, we get

‖S(p)n ( f ) − f (p)

‖r ≥1n

{∥∥∥∥[e1

2f ′′

](p)∥∥∥∥

r−

1n

∥∥∥∥∥ p!

2π

∫Γ

n2(Sn( f )(v) − f (v) −

v2n f ′′(v)

)(v − e1)p+1 dv

∥∥∥∥∥r

},

which combined again with Theorem 3.2 and taking into account that∥∥∥[ e1

2 f ′′](p)

∥∥∥r

> 0 (since f is not a polynomial

of degree ≤ p), as above leads us to the same conclusion. �

3. Voronovskaja’s theorem

In the case of real Favard–Szasz–Mirakjan operators attached to a function f : [0, ∞) → R, the followingVoronovskaja-type result is known.

Theorem 3.1 (Pop [8]). If f : [0, +∞) → R is twice continuous differentiable on [0, +∞), then

limn→∞

n[Sn( f )(x) − f (x)] =x

2f ′′(x),

uniformly in any compact subinterval of [0, ∞).

In this section we extend this result to the complex Favard–Szasz–Mirakjan operator, attached to a complex functionof the type in Theorem 2.1, obtaining in addition a quantitative estimate too.

We present

Theorem 3.2. Suppose that the hypothesis on the function f and the constants R, M, C, B, A in the statement ofTheorem 2.1 hold and let 1 ≤ r < 1

A be arbitrary fixed. The following Voronovskaja-type result holds∣∣∣Sn( f )(z) − f (z) −z

2nf ′′(z)

∣∣∣ ≤3M A|z|

r2n2

∞∑k=2

(k + 1)(r A)k−1, for all n ∈ N, |z| ≤ r.

Proof. Denoting ek(z) = zk , k = 0, 1, . . ., and Tn,k(z) = Sn(ek)(z), by the proof of Theorem 2.1, (i), we can writeSn( f )(z) =

∑∞

k=0 ck Tn,k(z), which immediately implies∣∣∣Sn( f )(z) − f (z) −z

2nf ′′(z)

∣∣∣ ≤

∞∑k=2

|ck | ·

∣∣∣∣Tn,k(z) − ek(z) −zk−1k(k − 1)

2n

∣∣∣∣for all z ∈ DR , n ∈ N.

S.G. Gal / Computers and Mathematics with Applications 56 (2008) 1121–1127 1125

By the recurrence relationship in the proof of Theorem 2.1, (i), satisfied by Tn,k(z), denoting Ek,n(z) = Tn,k(z) −

ek(z) −zk−1k(k−1)

2n , we immediately obtain the new recurrence

Ek,n(z) =z

nE ′

k−1,n(z) + zEk−1,n(z) +zk−2(k − 1)(k − 2)2

2n2 ,

for all k ≥ 2, n ∈ N and z ∈ DR .This implies, for all |z| ≤ r , k ≥ 2, n ∈ N,

|Ek,n(z)| ≤|z|

2n[2‖E ′

k−1,n‖r ] + |z| · |Ek−1,n(z)| +|z|

2n·

rk−3(k − 1)(k − 2)2

n

≤ r |Ek−1,n(z)| +|z|

2n

[2‖E ′

k−1,n‖r +rk−3(k − 1)(k − 2)2

n

]≤ r |Ek−1,n(z)| +

|z|

2n

[2(k − 1)

r‖Ek−1,n‖r +

rk−3(k − 1)(k − 2)2

n

]≤ r |Ek−1,n(z)|

+|z|

2n

[2(k − 1)

r‖Tn,k−1 − ek−1‖r +

2(k − 1)

r·

rk−2(k − 1)(k − 2)

2n+

rk−3(k − 1)(k − 2)2

n

]≤ r |Ek−1,n(z)| +

|z|

2n

[2(k − 1)

r·

rk−2k!

2n+

2(k − 1)

r·

rk−2(k − 1)(k − 2)

2n+

rk−3(k − 1)(k − 2)2

n

]≤ r |Ek−1,n(z)| +

3|z|rk−3

2n2 (k − 1)k! ≤ r |Ek−1,n(z)| +3|z|rk−3

2n2 (k + 1)!,

that is

|Ek,n(z)| ≤ r |Ek−1,n(z)| +3|z|rk−3

2n2 (k + 1)!, for all |z| ≤ r.

Taking k = 2, 3, . . . , in this last inequality, step by step we obtain

|Ek,n(z)| ≤3|z|rk−3

2n2

k+1∑j=3

j ! ≤3|z|rk−3(k + 1)!

n2 ,

which implies∣∣∣Sn( f )(z) − f (z) −z

2nf ′′(z)

∣∣∣ ≤

∞∑k=2

|ck | · |Ek,n(z)|

≤3M |z|

n2

∞∑k=2

Ak

k!(k + 1)!rk−3

≤3M A|z|

r2n2

∞∑k=2

(k + 1)(r A)k−1, for all |z| ≤ r,

where for r A < 1 we obviously have∑

∞

k=2(k + 1)(r A)k−1 < ∞. �

Remarks. (1) In the hypothesis on f in Theorem 2.1, it follows that limn→∞ n[Sn( f )(z)− f (z)] =z f ′′(z)

2 , uniformlyin any compact disk included in the open disk of center 0 and radius R.

(2) For functions f : D1 → C, analytic in D1 and continuous in D1, it is natural to attach other Bernstein-typepolynomials too, as follows (here i2 = −1):

Qn( f )(z) =

n∑k=0

(n

k

)zk(1 − z)n−k f (e

2kπ in ), z ∈ D1

or of the form

Pn( f )(z) =

n∑k=0

(n

k

)zk(1 − z)n−k f (e

kπ in ), z ∈ D1.

1126 S.G. Gal / Computers and Mathematics with Applications 56 (2008) 1121–1127

It would be of interest to study the approximation properties and Voronovskaja-type results for these complexpolynomials too.

4. Shape-preserving properties

The main result of this section is the following.

Theorem 4.1. Suppose that the hypothesis on the function f and the constants R, M, C, B, A in the statement ofTheorem 2.1 hold and let γ ∈ (−π/2, π/2).

If f (0) = f ′(0) − 1 = 0 and f is starlike (convex, spirallike of type γ , respectively) in D1, that is for all z ∈ D1(see e.g. [9])

Re(

z f ′(z)

f (z)

)> 0

(Re

(z f ′′(z)

f ′(z)

)+ 1 > 0, Re

(eiγ z f ′(z)

f (z)

)> 0, resp.

), (1)

then there exists an index n0 depending on f (and on γ for spirallikeness), such that for all n ≥ n0, the image of fby the complex Favard–Szasz–Mirakjan operators Sn( f )(z), are starlike (convex, spirallike of type γ , respectively) inD1.

Proof. Suppose first that f is starlike in D1. By Theorem 2.1, (ii), it follows that for n → ∞, we have Sn( f )(z) →

f (z), S′n( f )(z) → f ′(z) and S′′

n ( f )(z) → f ′′(z), uniformly in D1. In all what follows, denote Pn( f )(z) =Sn( f )(z)n f (1/n)

.Since Sn( f )(0) = f (0) and S′

n( f )(0) = n[ f (1/n) − f (0)], by f (0) = f ′(0) − 1 = 0 and the univalence of

f , we obtain n f (1/n) 6= 0, Pn( f )(0) =f (0)

n f (1/n)= 0, P ′

n( f )(0) =S′

n( f )(0)

n f (1/n)= 1, n ≥ 2, n f (1/n) =

f (1/n)− f (0)1/n

converges to f ′(0) = 1 as n → ∞, which means that for n → ∞, we have Pn( f )(z) → f (z), P ′n( f )(z) → f ′(z)

and P ′′n ( f )(z) → f ′′(z), uniformly in D1.

Then, by hypothesis we obtain | f (z)| > 0 for all z ∈ D1 with z 6= 0, which from the univalence of f in D1, impliesthat we can write f (z) = zg(z), with g(z) 6= 0, for all z ∈ D1, where g is analytic in D1 and continuous in D1.

Writing Pn( f )(z) in the form Pn( f )(z) = zQn( f )(z), obviously Qn( f )(z) is a polynomial of degree ≤ n − 1.Also, for |z| = 1 we have

| f (z) − Pn( f )(z)| = |z| · |g(z) − Qn( f )(z)| = |g(z) − Qn( f )(z)|,

which by the uniform convergence in D1 of Pn( f ) to f and by the maximum modulus principle, implies the uniformconvergence in D1 of Qn( f )(z) to g(z).

Since g is continuous in D1 and |g(z)| > 0 for all z ∈ D1, there exist an index n1 ∈ N and a > 0 depending on g,such that |Qn( f )(z)| > a > 0, for all z ∈ D1 and all n ≥ n1. Also, for all |z| = 1, we have

| f ′(z) − P ′n( f )(z)| = |z[g′(z) − Q′

n( f )(z)] + [g(z) − Qn( f )(z)]|

≥ | |z| · |g′(z) − Q′n( f )(z)| − |g(z) − Qn( f )(z)| |

= | |g′(z) − Q′n( f )(z)| − |g(z) − Qn( f )(z)| |,

which from the maximum modulus principle, the uniform convergence of P ′n( f ) to f ′ and of Qn( f ) to g, evidently

implies the uniform convergence of Q′n( f ) to g′.

Then, for |z| = 1, we get

z P ′n( f )(z)

Pn( f )(z)=

z[zQ′n( f )(z) + Qn( f )(z)]

zQn( f )(z)

=zQ′

n( f )(z) + Qn( f )(z)

Qn( f )(z)→

zg′(z) + g(z)

g(z)=

f ′(z)

g(z)=

z f ′(z)

f (z),

which again from the maximum modulus principle, implies

z P ′n( f )(z)

Pn( f )(z)→

z f ′(z)

f (z), uniformly in D1.

S.G. Gal / Computers and Mathematics with Applications 56 (2008) 1121–1127 1127

Since Re(

z f ′(z)f (z)

)is continuous in D1, there exists α ∈ (0, 1), such that

Re(

z f ′(z)

f (z)

)≥ α, for all z ∈ D1.

Therefore

Re[

z P ′n( f )(z)

Pn( f )(z)

]→ Re

[z f ′(z)

f (z)

]≥ α > 0

uniformly on D1, i.e. for any 0 < β < α, there is n0 such that for all n ≥ n0 we have

Re[

z P ′n( f )(z)

Pn( f )(z)

]> β > 0, for all z ∈ D1.

Since Pn( f )(z) differs from Sn( f )(z) only by a constant, this proves the starlikeness of Sn( f )(z).The proofs in the cases when f is convex or spirallike of order γ are similar and follow from the following uniform

convergences (on D1 or on Dr )

Re[

z P ′′n ( f )(z)

P ′n( f )(z)

]+ 1 → Re

[z f ′′(z)

f ′(z)

]+ 1. (2)

and

Re[

eiγ z P ′n( f )(z)

Pn( f )(z)

]→ Re

[eiγ z f ′(z)

f (z)

]. � (3)

Remarks. (1) Replacing D1 in the reasonings of Theorem 4.1 by Dr we immediately obtain that if f (0) = f ′(0)−1 =

0 and f is starlike (convex, spirallike of type γ , respectively) only in D1 (that is the corresponding inequalities holdonly in D1), then for any disk of radius 0 < r < 1 and center 0 denoted by Dr , there exists an index n0 = n0( f, Dr )

(n0 depends on γ too in the case of spirallikeness), such that for all n ≥ n0, the image of f by the complexFavard–Szasz–Mirakjan operators Sn( f )(z), are starlike (convex, spirallike of type γ , respectively) in Dr (that is,the corresponding inequalities hold in Dr ).

(2) The methods in this paper could also be used to other Bernstein-type operators appearing inapproximation theory, as for example to the Durrmeyer-type polynomials, Kantorovich-type polynomials, Stancu-type polynomials and for the non-polynomial operators of Baskakov, Meyer-Konig–Zeller, Jakimovski–Leviatan,Bleimann–Butzer–Hahn, Bernstein–Kantorivich, of Gamma-type, etc.

Note that, for example in [10], the uniform convergence of the complex generalized Bernstein polynomials ofJakimovski and Leviatan is proved. However, quantitative estimates of the approximation errors, Voronovskaja’stheorems and shape-preserving properties for this complex operator still remain to be studied.

Acknowledgements

The author would like to thank the referees for their valuable remarks. Paper supported by the Romanian Ministryof Education and Research, under CEEX grant, code 2-CEx 06-11-96.

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