# On the subordination under Bernardi operator.

1. Introduction. Let H denote the class of analytic functions in the unit disc U = {z: [absolute value of z < 1} on the complex plane C. For a [member of] C and n [member of] N we denote by

H[a,n] = {f [member of] H: f(z) = a + [a.sub.n][z.sup.n] + ...}

and

[A.sub.n] = {f [member of] H: f(z)= z + [a.sub.n+1][z.sup.n+1] + ...},

so A = [A.sub.1]. Let S be the subclass of A whose members are univalent in U.

The class [S.sup.*.sub.[alpha]] of starlike functions of order [alpha] < 1 may be defined as

[S.sup.*.sub.[alpha]] = {f [member of] A: [Real part]e[zf'(z)/f(z)] > [alpha], z [member of] U}.

The class [S.sup.*.sub.[alpha]] and the class [K.sub.[alpha]] of convex functions of order [alpha] < 1

[K.sub.[alpha]]: = {f [member of] A: [Real part](1 + [zf"(z)/f'(z)]) > [alpha], z [member of] U}

= {f [member of] A: zf' [member of] [S.sup.*.sub.[alpha]]}

were introduced by Robertson in . If [alpha] [member of] [0,1), then a function in either of these sets is univalent, if [alpha] < 0 it may fail to be univalent. In particular we denote [S.sup.*.sub.0] = [S.sup.*], [K.sub.0] = K, the classes of starlike and convex functions, respectively. Recall that f [member of] A is said to be in the class [C.sub.[alpha]], , of close-to-convex functions of order [alpha], [alpha] < 1, if and only if there exist g [member of] [S.sup.*.sub.[alpha]], [phi] [member of] R, such that

[Real part]e[e.sup.i[phi]][zf'(z)/g(z)] > 0, z [member of] U.

For f(z) = [a.sub.0] + [a.sub.1]z + [a.sub.2][z.sup.2] + ... and g(z)= [b.sub.0] + [b.sub.1]z + [b.sub.2][z.sup.2] + ... the Hadamard product (or convolution) is defined by (f * g)(z) = [a.sub.0][b.sub.0] + [a.sub.1][b.sub.1]z + [a.sub.2][b.sub.2][z.sup.2] + .... If X,Y [subset] H we also use the notation

X * Y: = {f * g: f [member of] X, g [member of] Y}.

The convolution has the algebraic properties of ordinary multiplication. The class A of analytic functions is closed under convolution, that is A * A = A. In 1973, Rusheweyh and Sheil-Small  proved the Polya-Schoenberg conjecture that the class of convex functions is preserved under convolution: K x K = K. Many other convolution problems were studied by St. Rusheweyh in  and have found many applications in various fields. We say that the f [member of] H is subordinate to g [member of] H in the unit disc U, written f [??] g if and only if there exits an analytic function w [member of] H such that w(0) = 0, [absolute value of w(z)] < 1 and f(z)= g[w(z)] for z [member of] U. Therefore, f [??] g in U implies f(U) [subset] g(U). In particular if g is univalent in U, then

(1.1) f [??] g, [f(0) = g(0) and f(U) [subset] g(U)].

2. Main result. The Alexander integral operator is defined by

A: [A.sub.n] [right arrow] [A.sub.n], A[f](z)= [[integral].sup.z.sub.0][f(t)/t] dt,

while

L: H [right arrow] H, L[f](z)= [2/z][[integral].sup.z.sub.0]f(t) dt

is the Libera operator . The above operators A and L are the special cases of the Bernardi operator  which is defined for k = 0 and for k [member of] C, [Real part]e{k} > 0, by

[L.sub.k]: H [right arrow] H, [L.sub.k][f](z)= [1 + k/[z.sup.k]][[integral].sup.z.sub.0]f(t)[t.sup.k-1] dt.

It is easy to see that

[L.sub.k]: [A.sub.n] [right arrow] [A.sub.n], [L.sub.k]: H[a,n] [right arrow] H[a(1 + k)/k, n].

Using the convolution we can write for f [member of] H[a, n]

(2.1) [L.sub.k][f](z) = f(z) * [[infinity].summation over (n=0)][k + 1]/[k + n][z.sup.n].

The classes [S.sup.*] and K are preserved under each of these operators whenever [Real part]e{k} > 0, Ruscheweyh  (earlier Bernardi  if k is a positive integer), i.e.: [L.sub.k][K] [subset] K, [L.sub.k][[S.sup.*]] [subset] [S.sup.*].

We shall need the following lemma.

Lemma 2.1 ([6, p. 35]). Suppose that the function [PSI]: [C.sup.2] x U [right arrow] C satisfies the condition [Real part]e{[PSI](i[??], [sigma])} [less than or equal to] [delta] for real [??],[sigma] [less than or equal to] -n(1 + [[??].sup.2])/2 and all z [member of] U. If q(z) = 1 + [a.sub.n][z.sup.n] + ... is analytic in U and

[Real part]e{[PSI](q(z), zq'(z))} > [delta]

for z [member of] Uy, then [Real part]e{q(z)} > 0 in U.

We note that Lemma 2.1 is a corollary of the fundamental result in theory of differential subordinations deeply developed by Miller and Mocanu . The function [PSI] is called admissible function.

Theorem 2.2. Let f be in the class [A.sub.n] and k be a non-negative real number. If

(2.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for z [member of] U, then [L.sub.k][f] is convex univalent function.

Proof. After some calculation we obtain

(2.3) q(z) + [zq'(z)/[k + q(z)]] = 1 + zf"(z)/f'(z),

where

(2.4) q(z) = 1 + z([L.sub.k][f](z))"/([L.sub.k][f](z))'.

It is known that [L.sub.k]: [A.sub.n] [right arrow] [A.sub.n], thus [L.sub.k][f] is of the form [L.sub.k][f](z)= z + [a.sub.n+1][z.sup.n+1] + .... If q is of the form q(z) = 1 + [c.sub.1]z + [c.sub.2][z.sup.2] + ..., then differentiating

z([L.sub.k][f](z))" = (q(z) - 1)([L.sub.k][f](z))'

and comparing the coefficients of both sides we obtain one after the other

[c.sub.1] = [c.sub.2] = ... = [c.sub.n-1] = 0, [c.sub.n] = n(n + 1)[a.sub.n+1],....

Therefore, q(z) = 1 + n(n + 1)[a.sub.n+1][z.sup.n] + .... To make use of Lemma 2.1 we consider the function

[PSI](r,s) = r + [s/k + r]

and [delta] = [[delta].sub.0](k). Then by (2.2), (2.3) we have [Real part]e{[PSI](q(z), zq'(z))} > [delta], furthermore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If [sigma] [less than or equal to] -n(1 + [[??].sup.2])/2, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Applying Lemma 2.1 with we obtain that [Real part]e{q(z)} > 0 for z [member of] U, hence trough (2.4) we see that [L.sub.k][f] is the convex univalent function whenever f satisfies (2.2). []

The above theorem is a generalization of the following one which is obtained from Theorem 2.2 with k = n = 1.

Corollary 2.3 ([6, p. 66]). Let f be in the class A. If

[Real part]e{1 + [zf"(z)/f'(z)]} > -1/2

for z [member of] U, then the function

L[f](z) = [2/z][[integral].sup.z.sub.0]f(t) dt

is in the class K of convex univalent functions.

The above property of the Libera operator L extends an earlier result in  that L[K] [subset] K. Note that the operator L is well defined in the whole class H.

Corollary 2.4. Let f be in the class [A.sub.n] and let k be a non-negative real number. Assume also that f satisfies condition (2.2). Then we have

(i) If g [member of] [C.sub.[alpha]] then [L.sub.k][g * f] [member of] [C.sub.[alpha]],

(ii) If g [member of] [S.sup.*.sub.[alpha]] then [L.sub.k][g * f] [member of] [S.sup.*.sub.[alpha]],

(iii) If g [member of] [K.sub.[alpha]] then [L.sub.k][g * f] [member of] [K.sub.[alpha]].

Proof. It is known , that the classes [C.sub.[alpha]], [S.sup.*.sub.[alpha]] and [K.sub.[alpha]] are closed under convolution with convex univalent and normalized functions. Because [L.sub.k][g * f] = g * [L.sub.k][f] and by Theorem 2.2 [L.sub.k][f] [member of] K the results (i)-(iii) becomes obvious.

Corollary 2.5. Let f be in the class S and let k be a non-negative real number. If r > 0 satisfies

[[r.sup.2] - 4r + 1]/[1 - [r.sup.2]] [greater than or equal to] [[delta].sub.0](k)

with [[delta].sub.0](k) given in (2.2), then [L.sub.k][f] is convex univalent in the disc [absolute value of z] < r.

Proof. It is known that f [member of] S, then for z = [re.sup.it]

[Real part]e{1 + [zf"(z)/f'(z)]} > [[r.sup.2] - 4r + 1]/[1 - [r.sup.2]].

Therefore, by Theorem 2.2 the function [L.sub.k][f] is convex univalent in the disc [absolute value of z] < r. []

We have [r.sup.2] - 4r + 1 > 0 for 0 [less than or equal to] r < 2 - [square root of 3] [approximately equal to] 0.2679, while [[delta].sub.0](k) [less than or equal to] 0. Therefore, if f [member of] S, k is a non-negative real number, and 0 [less than or equal to] r < 2 - [square root of 3], then [L.sub.k][f] is convex univalent in the disc [absolute value of z] < r. The above corollary for the Koebe function f(z) = z/[(1 - z).sup.2] and k = 1 becomes the following one.

Corollary 2.6. The function

[L.sub.1][z/[(1 - z).sup.2]] = 2{[1/[1 - z]] + [1/z]log(1 - z)}

= [[infinity].summation over (n=1)][2n/[n + 1]][z.sup.n]

is convex univalent in the disc [absolute value of z] < 4 - [square root of 13] [approximately equal to] 0.39.

Corollary 2.7. Let h be in the class [A.sub.n] and k be a non-negative real number. Assume that

(2.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for z [member of] U. Assume also that g(z) = a + [b.sub.n][z.sup.n] + [b.sub.n+1][z.sup.n+1] + ... is analytic in U. If

(2.6) g(z) + [zg'(z)/c] [??] [L.sub.k][h] (z [member of] U)

for [Real part]e[c] [greater than or equal to] 0, c [not equal to] 0, then

(2.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Moreover, the function [q.sub.n](z) is convex univalent and is the best dominant of (2.6) in the sense that g [??] [q.sub.n] for all g satisfying (2.6), and if there exists q such that g [??] q for all g satisfying (2.6), then [q.sub.n] [??] q.

Proof. It is known  that the subordination (2.6) with convex univalent right-hand side is sufficient for (2.7) with the best dominant [q.sub.n](z). By Theorem 2.2 the function [L.sub.k][h] is convex univalent in the unit disc and we get the result.

Notice that the function [q.sub.n](z) is the Bernardi integral operator on the function [L.sub.k][h]:

[q.sub.n](z) = [1/[1 + n]][L.sub.c/n][[L.sub.k][h] - a](z) + a.

Theorem 2.8. Assume that k is a complex number with [Real part]e{k} > 0, or k = 0. If g [member of] H and f is in the class [S.sup.*] of starlike functions, then

(2.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. The class [S.sup.*] is preserved under the operator [L.sub.k] whenever k = 0 or [Real part]{k} > 0, Ruscheweyh , i.e.: [L.sub.k][[S.sup.*]] [subset] [S.sup.*]. This fact was proved in  too. Note that if f [member of] S only, then [L.sub.k][f] may be infinite-valent in the unit disc. Because [L.sub.k][f] is univalent, then there exists a function w, w(0) = 0, such that in a disc [absolute value of z < [r.sub.0] [less than or equal to] 1

(2.9) [L.sub.k][g](z) = [L.sub.k][f](w(z)).

If [L.sub.k][g] [??} [L.sub.k][f], then there exists a [z.sub.0] [member of] U, such that [absolute value of w([z.sub.0])] = 1.

From (2.9) we have

[z.sup.k][L.sub.k][g](z) = [z.sup.k][L.sub.k][f](w(z)),

hence by (2.1)

(2.10) [z.sup.k]g(z) * [[infinity].summation over (n=1)][[k + 1]/[k + n]][z.sup.k+n]

= [z.sup.k]f(w(z)) * [[infinity].summation over (n=1)] [[k + 1]/[k + n]][z.sup.k+n].

The property z(p(z) * q(z))' = p(z) * zq'(z) used in (2.10) yields

(2.11) [z.sup.k]g(z) x [[infinity].summation over (n=1)](k + 1)[z.sup.k+n]

= [z.sup.k]f(w(z)) x [[infinity].summation over (n=1)](k + 1)[z.sup.k+n].

or, equivalently

(2.12) g(z) = f(w(z))

Because f is starlike univalent and there exists a [z.sub.0] [member of] U, such that [absolute value of w([z.sub.0])] = 1, we obtain a contradiction with g [??] f. []

Finally, we give the two applications of Theorem 2.2. If we consider for a [member of] [1,2] the function

(2.13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then [p.sub.a] [member of] [A.sub.1] and it satisfies

[Real part]e(1 + [[zp".sub.a](z)/[p'.sub.a](z)] = [Real part][[1 + az]/[1 - z]] > -[[a - 1]/z], z [member of] U,

thus [p.sub.a] satisfies condition (2.2) with k = a - 1 such that 0 [less than or equal to] k [less than or equal to] 1. Therefore, in this case, by Theorem 2.2 and by (2.1) the function

[L.sub.a-1][p.sub.a](z) = [p.sub.a](z) x [[infinity].summation over (n=0)][a/[a - 1 + n]][z.sup.n]

= [[infinity].summation over (n=0)][[(a).sub.n]/[a - 1 + n]n!][z.sup.n]

is convex univalent function.

Secondly, considering for l [member of] [1,2] the function

[r.sub.l](z) = [z/[(1 + [z.sup.l]).sup.1/l]] = z([[infinity].summation over (n=0)][[(1/l).sub.n]/n!][z.sup.ln], z [member of] U,

it is easy to check that [r.sub.l] [member of] [A.sub.1] and

[Real part]e(1 + [[zr".sub.l](z)/[r'.sub.l](z)]) = [1 - [lz.sup.l]]/[1 - [z.sup.l]] > -[l - 1]/z, z [member of] U

Therefore, [r.sub.l] satisfies condition (2.2) with k = l - 1 such that 0 [less than or equal to] k [less than or equal to] 1. By Theorem 2.2 the function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is convex univalent function.

doi: 10.3792/pjaa.89.11

Acknowledgment. The authors would like to express their sincerest thanks to the referees for a careful reading and various suggestions made for the improvement of the paper.

References

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 D.J. Hallenbeck and St. Ruscheweyh, Subordination by convex functions, Proc. Am. Math. Soc. 52 (1975), 191 195.

 W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J. 1 (1952), issue 2, 169 185.

 Z. Lewandowski, S. Miller and E. Zlotkiewicz, Generating functions for some classes of univalent functions, Proc. Am. Math. Soc. 56 (1976), 111 117.

 R. J. Libera, Some classes of regular univalent functions, Proc. Am. Math. Soc. 16 (1965), 755-758.

 S.S.Millerand P. T. Mocanu, Differential subordinations, Monographs and Textbooks in Pure and Applied Mathematics, 225, Dekker, New York, 2000.

 M. I. S. Robertson, On the theory of univalent functions, Ann. of Math. (2) 37 (1936), no. 2, 374 408.

 St. Ruscheweyh, New criteria for univalent functions, Proc. Am. Math. Soc. 49 (1975), 109-115.

 St. Ruscheweyh, Convolutions in geometric function theory, Sneminaire de Mathnematiques Supnerieures, 83, Presses Univ. Montreal, Montreal, QC, 1982.

 St. Ruscheweyh and T. Sheil-Small, Hadamard products of Schlicht functions and the Polya-Schoenberg conjecture, Comment.Math. Helv. 48 (1973), 119 135.

(Communicated by Kenji FUKAYA, M.J.A., Dec. 12, 2012)

2010 Mathematics Subject Classification. Primary 30C45; Secondary 30C80.

Janusz SOKOL, Department of Mathematics, RzeszowUniversity of Technology, Al. Powstancow Warszawy 12, 35-959 Rzeszow, Poland.

Mamoru NUNOKAWA, University of Gunma, Hoshikuki-cho 798-8, ChuouWard, Chiba 260-0808, Japan.
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