**1. Introduction**

Let <sup>H</sup> denote the class of analytic functions in the open unit disk <sup>U</sup> :<sup>=</sup> {*<sup>z</sup>* <sup>∈</sup> <sup>C</sup> : <sup>|</sup>*z*<sup>|</sup> <sup>&</sup>lt; <sup>1</sup>} and <sup>A</sup> denote the subclass of H consisting of functions of the form

$$f(z) = z + \sum\_{n=2}^{\infty} a\_n z^n. \tag{1}$$

Also, let <sup>S</sup> be the subclass of <sup>A</sup> consisting of all univalent functions in <sup>U</sup>. Then the logarithmic coefficients *γ<sup>n</sup>* of *f* ∈ S are defined with the following series expansion:

$$\log\left(\frac{f(z)}{z}\right) = 2\sum\_{n=1}^{\infty} \gamma\_n(f)z^n, \; z \in \mathbb{U}.\tag{2}$$

These coefficients play an important role for various estimates in the theory of univalent functions. Note that we use *γ<sup>n</sup>* instead of *γn*(*f*). The idea of studying the logarithmic coefficients helped Kayumov [1] to solve Brennan's conjecture for conformal mappings.

Recall that we can rewrite (2) in the series form as follows:

$$2\sum\_{n=1}^{\infty} \gamma\_n z^n = a\_2 z + a\_3 z^2 + a\_4 z^3 + \dots - \frac{1}{2} [a\_2 z + a\_3 z^2 + a\_4 z^3 + \dotsb]^2$$

$$+ \frac{1}{3} [a\_2 z + a\_3 z^2 + a\_4 z^3 + \dotsb]^3 + \dotsb$$

Now, considering the coefficients of *z<sup>n</sup>* for *n* = 1, 2, 3, it follows that

$$\begin{cases} \ 2\gamma\_1 = a\_{2, \prime} \\ \ 2\gamma\_2 = a\_3 - \frac{1}{2}a\_{2, \prime}^2 \\ \ 2\gamma\_3 = a\_4 - a\_2 a\_3 + \frac{1}{3}a\_2^3 \end{cases} \tag{3}$$

For two functions *f* and *g* that are analytic in U, we say that the function *f* is subordinate to *g* in <sup>U</sup> and write *<sup>f</sup>* (*z*) <sup>≺</sup> *<sup>g</sup>* (*z*) if there exists a Schwarz function *<sup>ω</sup>* that is analytic in <sup>U</sup> with *<sup>ω</sup>* (0) <sup>=</sup> 0 and |*ω* (*z*)| < 1 such that

$$f\left(z\right) = \mathfrak{g}\left(\omega\left(z\right)\right) \quad \left(z \in \mathbb{U}\right)\dots$$

In particular, if the function *<sup>g</sup>* is univalent in <sup>U</sup>, then *<sup>f</sup>* <sup>≺</sup> *<sup>g</sup>* if and only if *<sup>f</sup>*(0) = *<sup>g</sup>*(0) and *<sup>f</sup>*(U) <sup>⊆</sup> *<sup>g</sup>*(U).

Using subordination, different subclasses of starlike and convex functions were introduced by Ma and Minda [2], in which either of the quantity *z f* (*z*) *<sup>f</sup>*(*z*) or 1 <sup>+</sup> *z f* (*z*) *<sup>f</sup>* (*z*) is subordinate to a more general superordinate function. To this aim, they considered an analytic univalent function *ϕ* with positive real part in U. *ϕ*(U) is symmetric respecting the real axis and starlike considering *ϕ*(0) = 1 and *ϕ* (0) > 0. They defined the classes consisting of several well-known classes as follows:

$$\mathcal{S}^\*(\varrho) := \left\{ f \in \mathcal{S} : \frac{zf'(z)}{f(z)} \prec \varrho(z), \ z \in \mathbb{U} \right\},$$

and

$$\mathcal{K}(\mathfrak{q}) := \left\{ f \in \mathcal{S} : 1 + \frac{z f''(z)}{f'(z)} \prec \mathfrak{q}(z), \; z \in \mathcal{U} \right\} \;.$$

For example, the classes <sup>S</sup>∗(*ϕ*) and <sup>K</sup>(*ϕ*) reduce to the classes <sup>S</sup>∗[*A*, *<sup>B</sup>*] :<sup>=</sup> <sup>1</sup> <sup>+</sup> *Az* <sup>1</sup> <sup>+</sup> *Bz* and <sup>K</sup>[*A*, *<sup>B</sup>*] :<sup>=</sup> 1 + *Az* <sup>1</sup> <sup>+</sup> *Bz* of the well-known Janowski starlike and Janowski convex functions for <sup>−</sup><sup>1</sup> <sup>≤</sup> *<sup>B</sup>* <sup>&</sup>lt; *<sup>A</sup>* <sup>≤</sup> 1, respectively. By replacing *A* = 1 − 2*α* and *B* = −1 where 0 ≤ *α* < 1, we conclude the classes S∗(*α*) and K(*α*) of the starlike functions of order *α* and convex functions of order *α*, respectively. In particular, S<sup>∗</sup> := S∗(0) and K := K(0) are the class of starlike functions and of convex functions in the unit disk <sup>U</sup>, respectively. The Koebe function *<sup>k</sup>*(*z*) = *<sup>z</sup>*/(<sup>1</sup> <sup>−</sup> *<sup>z</sup>*)<sup>2</sup> is starlike but not convex in <sup>U</sup>. Thus, every convex function is starlike but not conversely; however, each starlike function is convex in the disk of radius 2 <sup>−</sup> <sup>√</sup>3.

Lately, several researchers have subsequently investigated similar problems in the direction of the logarithmic coefficients, the coefficient problems, and differential subordination [3–11], to mention a few. For example, the rotation of Koebe function *<sup>k</sup>*(*z*) = *<sup>z</sup>*(<sup>1</sup> <sup>−</sup> *<sup>e</sup>i<sup>θ</sup>* )−<sup>2</sup> for each *<sup>θ</sup>* has logarithmic coefficients *<sup>γ</sup><sup>n</sup>* <sup>=</sup> *<sup>e</sup>iθn*/*n*, *<sup>n</sup>* <sup>≥</sup> 1. If *<sup>f</sup>* ∈ S, then by using the Bieberbach inequality for the first equation of (3) it concludes |*γ*1| ≤ 1 and by utilizing the Fekete–Szegö inequality for the second equation of (3), (see [12] (Theorem 3.8)),

$$|\gamma\_2| = \frac{1}{2}|a\_3 - \frac{1}{2}a\_2^2| \le \frac{1}{2}(1 + 2\varepsilon^{-2}) = 0.635\dotsb \dotsb$$

It was shown in [12] (Theorem 4) that the logarithmic coefficients *γ<sup>n</sup>* of every function *f* ∈ S satisfy

$$\sum\_{n=1}^{\infty} |\gamma\_n|^2 \le \frac{\pi^2}{6},$$

and the equality is attained for the Koebe function. For *f* ∈ S∗, the inequality |*γn*| ≤ 1/*n* holds but is not true for the full class S, even in order of magnitude (see [12] (Theorem 8.4)). In 2018, Ali and Vasudevarao [3] and Pranav Kumar and Vasudevarao [6] obtained the logarithmic coefficients *γ<sup>n</sup>* for certain subclasses of close-to-convex functions. Nevertheless, the problem of the best upper bounds for the logarithmic coefficients of univalent functions for *n* ≥ 3 is presumably still a concern.

Based on the results presented in previous research, in the current study, the bounds for the logarithmic coefficients *γ<sup>n</sup>* of the general classes S∗(*ϕ*) and K(*ϕ*) were estimated. It is worthwhile mentioning that the given bounds in this paper would generalize some of the previous papers and that many new results are obtained, noting that our method is more general than those used by others. The following lemmas will be used in the proofs of our main results.

For this work, let Ω represent the class of all analytic functions *ω* in U that equips with conditions *<sup>ω</sup>*(0) = 0 and <sup>|</sup>*ω*(*z*)<sup>|</sup> <sup>&</sup>lt; 1 for *<sup>z</sup>* <sup>∈</sup> <sup>U</sup>. Such functions are called Schwarz functions.

**Lemma 1.** *[13] (p. 172) Assume that ω is a Schwarz function so that ω*(*z*) = ∑<sup>∞</sup> *<sup>n</sup>*=<sup>1</sup> *pnzn*. *Then*

$$|p\_1| \le 1, \quad |p\_n| \le 1 - |p\_1|^2 \quad n = 2, 3, \dots$$

**Lemma 2.** *[14] Let <sup>ψ</sup>*, ∈ H *be any convex univalent functions in* <sup>U</sup>*. If <sup>f</sup>*(*z*) <sup>≺</sup> *<sup>ψ</sup>*(*z*) *and <sup>g</sup>*(*z*) <sup>≺</sup> (*z*)*, then f*(*z*) ∗ *g*(*z*) ≺ *ψ*(*z*) ∗ (*z*) *where f* , *g* ∈ H*.*

We observe that in the above lemma, nothing is assumed about the normalization of *ψ* and , and "∗" represents the Hadamard (or convolution) product.

**Lemma 3.** *[12,15] (Theorem 6.3, p. 192; Rogosinski's Theorem II (i)) Let <sup>f</sup>*(*z*) = <sup>∞</sup> ∑ *n*=1 *anz<sup>n</sup> and g*(*z*) = ∞ ∑ *bnz<sup>n</sup> be analytic in* <sup>U</sup>*, and suppose that f* <sup>≺</sup> *g where g is univalent in* <sup>U</sup>*. Then*

$$\sum\_{k=1}^{n} \left| a\_k \right|^2 \le \sum\_{k=1}^{n} \left| b\_k \right|^2, \quad n = 1, 2, \dots$$

**Lemma 4.** *[12,15] (Theorem 6.4 (i), p. 195; Rogosinski's Theorem X) Let <sup>f</sup>*(*z*) = <sup>∞</sup> ∑ *n*=1 *anz<sup>n</sup> and g*(*z*) = ∞ ∑ *bnz<sup>n</sup> be analytic in* <sup>U</sup>*, and suppose that f* <sup>≺</sup> *g where g is univalent in* <sup>U</sup>*. Then*

*n*=1

*(i) If g is convex, then* |*an*|≤|*g* (0)| = |*b*1|, *n* = 1, 2, . . .*.*

*(ii) If g is starlike (starlike with respect to 0), then* |*an*| ≤ *n*|*g* (0)| = *n*|*b*1|, *n* = 2, 3, . . .*.*

**Lemma 5.** *[16] If ω*(*z*) = ∑<sup>∞</sup> *<sup>n</sup>*=<sup>1</sup> *pnz<sup>n</sup>* <sup>∈</sup> <sup>Ω</sup>*, then for any real numbers <sup>q</sup>*<sup>1</sup> *and <sup>q</sup>*2*, the following sharp estimate holds:*

$$|p\_3 + q\_1 p\_1 p\_2 + q\_2 p\_1^3| \le H(q\_1; q\_2)\_{\prime \prime}$$

*where*

*n*=1

$$H(q\_{1};q\_{2}) = \begin{cases} 1 & \text{if } \quad (q\_{1},q\_{2}) \in D\_{1} \cup D\_{2} \cup \{ (2,1) \}, \\ |q\_{2}| & \text{if } \quad (q\_{1},q\_{2}) \in \cup\_{k=3}^{2} D\_{k}, \\ \frac{2}{3}(|q\_{1}|+1) \left( \frac{|q\_{1}|+1}{3(|q\_{1}|+1+q\_{2})} \right)^{\frac{1}{2}} & \text{if } \quad (q\_{1},q\_{2}) \in D\_{8} \cup D\_{9}, \\ \frac{q\_{2}}{3} \left( \frac{q\_{1}^{2}-4}{q\_{1}^{2}-4q\_{2}} \right) \left( \frac{q\_{1}^{2}-4}{3(q\_{2}-1)} \right)^{\frac{1}{2}} & \text{if } \quad (q\_{1},q\_{2}) \in D\_{10} \cup D\_{11} \backslash \{ (2,1) \}, \\ \frac{2}{3}(|q\_{1}|-1) \left( \frac{|q\_{1}|-1}{3(|q\_{1}|-1-q\_{2})} \right)^{\frac{1}{2}} & \text{if } \quad (q\_{1},q\_{2}) \in D\_{12}. \end{cases}$$

*While the sets Dk*, *k* = 1, 2, . . . , 12 *are defined as follows:*

*D*<sup>1</sup> = . (*q*1, *<sup>q</sup>*2) : <sup>|</sup>*q*1| ≤ <sup>1</sup> 2 , |*q*2| ≤ 1 / , *D*<sup>2</sup> = . (*q*1, *<sup>q</sup>*2) : <sup>1</sup> <sup>2</sup> ≤ |*q*1| ≤ 2, <sup>4</sup> 27 (|*q*1<sup>|</sup> <sup>+</sup> <sup>1</sup>)<sup>3</sup> − (|*q*1| + 1) ≤ |*q*2| ≤ 1 / , *D*<sup>3</sup> = . (*q*1, *<sup>q</sup>*2) : <sup>|</sup>*q*1| ≤ <sup>1</sup> 2 , |*q*2|≤−1 / , *D*<sup>4</sup> = . (*q*1, *<sup>q</sup>*2) : <sup>|</sup>*q*1| ≥ <sup>1</sup> 2 , <sup>|</sup>*q*2|≤− <sup>2</sup> 3 (|*q*1| + 1) / , *D*<sup>5</sup> = {(*q*1, *q*2) : |*q*1| ≤ 2, |*q*2| ≥ 1} , *D*<sup>6</sup> = . (*q*1, *<sup>q</sup>*2) : 2 ≤ |*q*1| ≤ 4, <sup>|</sup>*q*2| ≥ <sup>1</sup> <sup>12</sup> (*q*<sup>2</sup> <sup>1</sup> + 8) / , *D*<sup>7</sup> = . (*q*1, *<sup>q</sup>*2) : <sup>|</sup>*q*1| ≥ 4, <sup>|</sup>*q*2| ≥ <sup>2</sup> 3 (|*q*1| − 1) / , *D*<sup>8</sup> = . (*q*1, *<sup>q</sup>*2) : <sup>1</sup> <sup>2</sup> ≤ |*q*1| ≤ 2, <sup>−</sup> <sup>2</sup> 3 (|*q*1<sup>|</sup> <sup>+</sup> <sup>1</sup>) <sup>≤</sup> *<sup>q</sup>*<sup>2</sup> <sup>≤</sup> <sup>4</sup> 27 (|*q*1<sup>|</sup> <sup>+</sup> <sup>1</sup>)<sup>3</sup> − (|*q*1| + 1) / , *D*<sup>9</sup> = (*q*1, *<sup>q</sup>*2) : <sup>|</sup>*q*1| ≥ 2, <sup>−</sup> <sup>2</sup> 3 (|*q*1<sup>|</sup> <sup>+</sup> <sup>1</sup>) <sup>≤</sup> *<sup>q</sup>*<sup>2</sup> <sup>≤</sup> <sup>2</sup>|*q*1|(|*q*<sup>1</sup> <sup>+</sup> <sup>1</sup>|) *q*2 <sup>1</sup> + 2|*q*1| + 4 3 , *D*<sup>10</sup> = (*q*1, *<sup>q</sup>*2) : 2 ≤ |*q*1| ≤ 4, <sup>2</sup>|*q*1|(|*q*<sup>1</sup> <sup>+</sup> <sup>1</sup>|) *q*2 <sup>1</sup> <sup>+</sup> <sup>2</sup>|*q*1<sup>|</sup> <sup>+</sup> <sup>4</sup> <sup>≤</sup> *<sup>q</sup>*<sup>2</sup> <sup>≤</sup> <sup>1</sup> <sup>12</sup> (*q*<sup>2</sup> <sup>1</sup> + 8) 3 , *D*<sup>11</sup> = (*q*1, *<sup>q</sup>*2) : <sup>|</sup>*q*1| ≥ 4, <sup>2</sup>|*q*1|(|*q*<sup>1</sup> <sup>+</sup> <sup>1</sup>|) *q*2 <sup>1</sup> <sup>+</sup> <sup>2</sup>|*q*1<sup>|</sup> <sup>+</sup> <sup>4</sup> <sup>≤</sup> *<sup>q</sup>*<sup>2</sup> <sup>≤</sup> <sup>2</sup>|*q*1|(|*q*<sup>1</sup> <sup>−</sup> <sup>1</sup>|) *q*2 <sup>1</sup> − 2|*q*1| + 4 3 , *D*<sup>12</sup> = (*q*1, *<sup>q</sup>*2) : <sup>|</sup>*q*1| ≥ 4, <sup>2</sup>|*q*1|(|*q*<sup>1</sup> <sup>−</sup> <sup>1</sup>|) *q*2 <sup>1</sup> <sup>−</sup> <sup>2</sup>|*q*1<sup>|</sup> <sup>+</sup> <sup>4</sup> <sup>≤</sup> *<sup>q</sup>*<sup>2</sup> <sup>≤</sup> <sup>2</sup> 3 (|*q*1| − 1) 3 .
