Geometric Properties of Partial Sums of Univalent Functions

Theorem 2.1 (Szeg¨o Theorem). The partial sums of univalent functions f ∈S are univalent in the disk D1/4 and the number 1/4 cannot be replaced by a larger one. Using an inequality of Goluzin, Jenkins [14] (as well as Ilieff[12], see Duren [4, §8.2, pp. 241–246]) found a simple proof of this result and also shown that the partial sums of odd univalent functions are univalent in D1/ √ 3. The number 1/ √ 3 is shown to be the radius of starlike- ness of the partial sums of the odd univalent function

Theorem 2.2. Let f ∈S and f 1/k(zk) = ∞ X ν=0 b(k) ν zνk+1, (k = 2, 3, . . . , b(k) 0 = 1.) Then Pn ν=0 b(k) ν zνk+1 are convex in D k√ k/(2(k+1)2). The radii of convexity are sharp.

Theorem 2.3. If f ∈clco S and g ∈F, then f ∗g is starlike in D1/4. The constant 1/4 is best possible. In particular, for g(z) = z+z2+· · ·+zn, Theorem 2.3 reduces to the following result.

Corollary 2.1. If f belongs to clco S or, in par- ticular, to the class of the normalized typically real func- tions, then the nth partial sum fn is starlike in D1/4. The constant 1/4 is best possible. The class F contains the following two subsets: R1/2 := {f ∈A : Re(f(z)/z) > 1/2, z ∈D} ⊂F and D := ( n X k=1 akzk ∈A : 0 ≤ak+1 ≤ak ) ⊂F. Since the class C of convex functions is a subset of R1/2,

Theorem 2.3 reduces to the following:

Corollary 2.2. If f belongs to F, then the function f and, in particular, the nth partial sum fn, is convex in D1/4. The constant 1/4 is best possible. We remark that Suffridge [48] has shown that the partial sums of the function e1+z are all convex. More generally, Ruscheweyh and Salinas [35] have shown that the functions of the form P∞ k=0 ak(1 + z)k/k!, a0 ≥a1 ≥ · · · ≥0 are either constant or convex univalent in the unit disk D. Let F(z) = z + P∞ 1 akz−k be analytic |z| > 1. Reade [27] obtain

Theorem 3.1. [29] (see [38, Theorem 2, p. 1193]) If f(z) = z + P∞ k=2 akzk is either starlike, or convex, or typically-real, or convex in the direction of imaginary axis, then there is n0 such that, for n ≥n0, the partial sum fn(z) := z + Pn k=2 akzk has the same property in Dρ where ρ ≥1 −3 log n/n. An analytic function f(z) = zp + P∞ k=1 ap+kzp+k is p-valently starlike [29, p. 830] if f assumes no value more than p times, at least one value p times and Re zf ′(z) f(z)

Theorem 3.2. If f is a starlike function, then every partial sum fn of f is convex in |z| < 1/8 and the number 1/8 cannot be increased. In view of the above theorem, the nth partial sum of Koebe function z/(1 −z)2 is convex in |z| < 1/8. A verification of this fact directly can be used to give another proof of this theorem by using the fact [34] that the convolution of two convex function is again convex. It is also known [34] that Re(f(z)/fn(z)) > 1/2 for a function f starlike of order 1/2. This

Theorem 3.3. If f ∈S∗(1/2), then Re  λzf ′(z) f(z) + µfn(z) f(z)  > 0 (z ∈D) provided that λ and µ are both nonnegative with at least one of them nonzero or provided that µ is a complex num- ber with |λ| > 4|µ|. The result is sharp in the sense that the ranges of λ and µ cannot be increased. 4. Partial sums of convex functions For a convex function f ∈C, it is well-known that

Theorem 4.1. If f ∈C, then the nth partial sum fn of f satisfies 1 −fn(z) f(z) ≤|z|n < 1 (z ∈D, n ≥1) and hence (4.1) Re f(z) fn(z) > 1 2 (z ∈D, n ≥1). As a consequence of this theorem, he has shown that the function Qn given by Qn(z) = R z

Theorem 4.2. If f is convex function, then every partial sum fn of f is convex in |z| < 1/4.

Theorem 4.3. [3, Theorem 4, p. 117] If f is convex, then the nth partial sum fn is starlike in |z| < rn where rn is the positive root of the equation 1 −(n + 1)rn − nrn+1 = 0. The result is sharp for each even n for f(z) = z/(1 −z). The above theorem with a weaker conclusion that fn is univalent was obtained earlier by Ruscheweyh [31]. Singh [44] proved that the nth partial sum fn of a star- like function of order 1/2 is starlike in |z| < rn where rn is given in Theorem 4.3. He has also shown th

Lemma 4.1. The function gn(z) = z(1−zn) 1−z is starlike of order α in |z| < rn where rn is the smallest positive root of the equation 1 −α −αr + (α −1 −n)rn + (α −n)rn+1 = 0. The result is sharp for even n.

Theorem 4.4. [38, Theorem 1, p. 1192] If f is con- vex, then the nth partial sum fn is starlike in |z| < (1/(2n))1/n for all n. In particular, fn is starlike in |z| < 1/2 and the radius 1/2 is sharp.

Lemma 4.2. The function gn(z) = z(1−zn) 1−z is in SP for |z| < rn where rn is the smallest positive root of the equation 1 −r = (1 + 2n)rn + (2n −1)rn+1. The result is sharp for even n.

Theorem 4.5. If f(z) is convex of order 1/2, then the partial sums fn are uniformly convex for |z| < rn where rn is the smallest positive root of 1 −r = (2n + 1)rn + (2n −1)rn+1.

Theorem 5.1. [21] If the analytic function f given by (1.1) satisfies the inequality |f ′(z)| ≤M, M > 1, then the radius of starlikeness of fn is 1/M.

Theorem 5.2. If the analytic function f given by (1.1) satisfies the inequality Re f ′(z) > 0, then fn is uni- valent in |z| < 1/2.

Theorem 5.3. [28] If p(z) = 1 + c1z + c2z2 + · · · is analytic and has positive real part in D, then, for n ≥2, pn(z) = 1 + c1z + c2z2 + · · · + cnzn has positive real part in Dρ where ρ is the root Rn ≥1 −2 log n/n in (0,1) of the equation (1 −r)2 = 2rn+1(1 + r). Singh [43] investigated the radius of convexity for functions whose derivative has positive real part and proved the following result.

Theorem 5.4. If the analytic function f given by (1.1) satisfies the inequality Re f ′(z) > 0, then fn is con- vex in |z| < 1/4. The number 1/4 cannot be replaced by a greater one. Extending Theorem 5.2 of MacGregor, Silverman [42] has shown that, whenever Re f ′(z) > 0, fn is univa- lent in {z : |z| < rn}, where rn is the smallest positive root of the equation 1 −r −2rn = 0, and the result is sharp for n even. He also shown that rn > (1/2n)1/n and rn > 1 −log n/n for n ≥5. Also he proved that th

Theorem 5.5. Let f ∈Rα. Then Re fn′(z) > 0 in the disc |z| < rn(α), where rn(α) is the least positive root of the equation 2rn+r−1+4αr/((1−α)(1+r)) = 0. Also fn is univalent for |z| < Rn(α), where Rn(α) is the least positive root of 2rn+r−1−α(1−r)2/((1−α)(1+r)) = 0. 6. Close-to-convex functions Recall that a function f ∈A satisfying the condition Re f ′(z) g′(z)  > 0 for some (not necessarily normalized) convex univalent function g, is called close-to-convex. In this section, some results rela

Theorem 6.1. [19] Let the analytic function f be given by (1.1). Let g(z) = z + b2z2 + · · · be convex. If Re(f ′(z)/g′(z)) > 0 for z ∈D, then Re(f ′ n(z)/g′ n(z)) > 0 for |z| < 1/4 and 1/4 is the best possible constant. The function f satisfying the hypothesis of the above theorem is clearly close-to-convex. This theorem implies that fn is also close-to-convex for |z| < 1/4 and therefore it is a generalization of Szeg¨o result. The result applies only to a subclass of the class of close-to-conv

Theorem 6.2. If f(z) = z + P∞ 2 aνzν is analytic and satisfy Re zf ′(z) φ(z) > 0, where φ(z) = z + P∞ 2 bνzν is starlike univalent, then, for each n > 1, Re (zf ′ n(z))′ φ′n(z) > 0 (|z| < 1 8), and the constant 1/8 cannot be replaced by any greater

Theorem 6.3. Let f(z) = z +P∞ 2 aνzν, be analytic and satisfy Re (zf ′(z))′ φ′(z) > 0, where φ(z) = z + P∞ 2 bνzν is schlicht and convex in |z| < 1. Then, for each n > 1, Re zf ′ n(z) φn(z) > 0 (|z| < 1 2). The constant 1/2 cannot be replaced by any greater one.

Theorem 6.4. [9] Let f(z) = z + P∞ 2 aνzν, be an- alytic and satisfy Re (zf ′(z))′ φ′(z) > 0, where φ(z) = z + P∞ 2 bνzν is starlike in |z| < 1. Then, for each n > 1, Re (zf ′ n(z))′ φ′n(z) > 0 (|z| < 1 6).

Theorem 6.5. If f(z) = P∞ k=1 akzk be linearly ac- cessible in D, then 1 −fn(z) f(z) ≤(2n + 1)|z|n (z ∈D).

Theorem 7.1. If the analytic function f satisfies ∞ X k=2 (k −α)|ak| ≤(1 −α) for some 0 ≤α < 1, then Re f(z) fn(z) ≥ n n + 1 −α, Re fn(z) f(z) ≥n + 1 −α n + 2 −2α, Re f ′(z) f ′n(z) ≥

Theorem 7.2. Let f ∈A, |a2| = 2b, 0 ≤b ≤1 and |an| ≤n for n ≥3. Then f satisfies the inequality

Theorem 8.1. If f ∈S has the form (8.1) z f(z) = 1 + b1z + b2z2 + · · · such that bk is real and non-negative for each k ≥2, then for each n ≥2

Theorem 8.2. If f ∈U has the form (8.1), then (8.3) ∞ X n=2 (n −1)2|bn|2 ≤1. In particular, we have |b1| ≤2 and |bn| ≤ 1 n−1 for n ≥2. The results are sharp.

Theorem 8.3. Suppose that f ∈U and fn(z) is its partial sum. Then for each n ≥2

Corollary 8.1. Suppose that f ∈U. Then for n ≥ 3 one has

Theorem 8.4. If f(z) = z+P∞ k=3 akzk (i.e. a2 = 0) belongs to the class U, then the n-th partial sum fn is in the class U in the disk |z| < r, where r is the unique positive root of the equation (1 −r)3(1 + r)2 −rn(1 + r2)2[5 + r + n(1 −r2)] = 0. In particular, for n ≥5, we have r ≥rn = 1 −3 log n −log(log n) n . For n = 3, 4, 5, one has r = 0.361697, r = 0.423274, r = 0.470298, respectively.

Theorem 8.5. Let f(z) = z +P∞ k=3 akzk (i.e. a2 = 0) belong to the class U. Then for each integer n ≥2, we have Re  f(z) fn(z)  > 1 2 in the disk |z| < p√ 5 −2. 9. Generalized Partial Sum By making use of the fact that the convolution of

Theorem 9.1. If f(z) = z + P∞ k=2 akzk is convex, then Fk(z) = z + P∞ j=1 ajk+1zjk+1, (k = 2, 3, . . . ), is starlike in |z| < (1/(k −1))1/k. The bound is sharp for every k. The proof follows from the following inequality sat- isfied by Gk(z) = z/(1 −zk): Re zG′ k(z) Gk(z) ≥1 −(k −2)rk −(k −1)r2k |1 −zk|2 (|z| = r < 1). Since (1/(k −1))1/k attains its minimum when k = 5, it follows that, for a convex function f, the Fk is starlike

Theorem 9.2. If f is convex, then the generalized partial sum ˜f of the function f is (1) convex univalent in |z| < c where c (≈0.20936) is the unique root in (0, 1) of the equation x(1 + x2)/(1 −x2)3 = 1/4. (2) starlike univalent in |z| < b where b (≈0.3715) is the unique root in (0, 1) of the equation x/(1 −x2)2 = 1/2. The function z + P∞ k=1 z2k = z + z2/(1 −z2) associated with the convex function z + P∞ k=2 zk = z/(1 −z) is extremal for the radii of convexity and starlikeness. These results

Theorem 9.3. If f ∈S, then the generalized partial sum ˜f of the function f satisfies Re ˜f ′(cz) > 0 for all z ∈ D, where c is as in Theorem 9.2. The function f(z) = z/(1 −z)2 and {nk}∞ k=2 = {2k −2}∞ k=2 show that the result is sharp. They [6] have also proved that if f is analytic and Re{f(z)/z} > 1 2, then |z ˜f ′′(z)| ≤Re ˜f ′(z) (|z| < c) for any choice of {nk}∞ k=2. For the class R of functions f in A for which Re(f ′(z)+

Theorem 9.4. Let r0 denote the positive root of the equation r +log(1−r2) = 0. If f ∈R, then Re ˜f ′(z) ≥0 for |z| ≤r0 ≈0.71455. The result is sharp, with extremal function ˜f(z) = z + 2 P∞ n=1 z2n/(2n)2. For functions f ∈R, it is also known [47] that the nth partial sum fn of f satisfies Re f ′ n(z) > 0 and hence fn is univalent. Also Re(fn(z)/z) > 1/3. Acknowledgement The author is thankful to Sumit Nagpal for carefully reading this manuscript. References [1] R. M. Ali, V. Ravichandran, Uniform