Although many investors like to focus on some of the new market opportunities that Apple (NASDAQ:AAPL) has ahead of it, such as Apple Watch and its services business, there's still plenty of opportunity for the company to grow its largest and most important business: iPhone.

On Apple's most recent earnings call, one analyst -- seemingly in an attempt to gauge Apple's view of this opportunity -- asked the company's management about the potential for smartphone market share gains.

Apple's iPad and iPhone next to one another.

Image source: Apple.

Apple CEO Tim Cook declined to give any specific guidance on Apple's market share plans, but he did indicate that he does expect the company to be able to grow its iPhone market share out in time.

Here's how Apple can do it, what kind of market share Apple could reasonably expect to gain, and what such gains could ultimately mean for the company.

Samsung's success is Apple's opportunity

A new report from Strategy Analytics (via GizmoChina) claims that during the second calendar quarter of the year, Samsung (NASDAQOTH:SSNLF) shipped 10.2 million Galaxy S8 smartphones and 9 million Galaxy S8+ smartphones, for a total of 19.2 million phones.

Apple's iPhone 7 and iPhone 7 Plus maintained sizable leads over their Samsung-made counterparts, shipping 16.9 million and 15.1 million units, per the report -- an impressive showing considering that the Galaxy S8-series launched in the second quarter of 2017 while the iPhone 7-series came out at the end of the third quarter of 2016.

I strongly suspect that Apple has an opportunity to eat into Samsung's flagship Galaxy S-series smartphone shipments beginning with this year's iPhone models -- and beyond. Here's how.

Apple can pull discerning customers away

Smartphone aesthetics and -- at least the perception of -- technological superiority both matter to a portion of the smartphone buying public. I suspect that this is particularly true among those that purchase flagship smartphones like the iPhone and the Galaxy S-series of devices.

Over the last two generations -- that is, during the iPhone 6s and the iPhone 7 generations -- Apple has fielded phones that, if we're being honest, looked nearly identical to the iPhone 6-series that launched in 2014.

By contrast, Samsung has been making larger strides in the form factor/aesthetics of its devices. The Galaxy S5 looked rather cheap compared to, say, the iPhone 5s, but beginning with the Galaxy S6, Samsung arguably began to pull ahead in terms of aesthetics.

The Galaxy S6 featured a glass back (which some think looks more "premium" than the aluminum unibody of the iPhone 6/6s/7-series devices), the Galaxy S7 came in an "Edge" variant with an edge-to-edge display that wrapped around the sides.

Then, with the Galaxy S8, Samsung introduced a display that not only retained that slick curved edge-to-edge display, but it introduced a "full face" display as well that nearly eliminated the bezels.

With this year's premium OLED iPhone, Apple has a real opportunity to pull over the kinds of customers that value aesthetics, as the device is rumored to adopt a glass back as well as a nearly bezel-free full-face OLED design.

Apple is rumored to have tested out a curved display, but that simply wasn't viable for this year's iPhone models. However, if Apple can deliver on a sharp, beautiful, and potentially even fast (with ProMotion) OLED display this year, then Apple will not only have "caught up" with Samsung (and others), but it'll be in the pole position vis-à-vis aesthetics.

That aesthetic leadership, coupled with technology leadership, brand strength, and the robustness of its software and services ecosystem, should allow Apple to capture significant share of the high-end smartphone market from Samsung and others.

Then, once Apple has established such leadership, should it continue to innovate faster and, frankly, better than others (something that Apple's huge research and development budget is clearly dedicated to doing), then it should be looking at years of sustained premium smartphone market share growth.

Ashraf Eassa has no position in any stocks mentioned. The Motley Fool owns shares of and recommends Apple. The Motley Fool has a disclosure policy.