# 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 [7]. 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]], [3], 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 [10] 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 [9] 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 [5]. The above operators A and L are the special cases of the Bernardi operator [1] 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 [8] (earlier Bernardi [1] 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 [6]. 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 [5] 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 [10], 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 [2] 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 [8], i.e.: [L.sub.k][[S.sup.*]] [subset] [S.sup.*]. This fact was proved in [4] 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

[1] S. D. Bernardi, Convex and starlike univalent functions, Trans. Am. Math. Soc. 135 (1969), 429 446.

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

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

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

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

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

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

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

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

[10] 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.