UPDATE | TL;DR :
I actually found shorter way:
    auto library = dlopen("/path/to/lib.so", RTLD_LAZY | RTLD_GLOBAL);
    const char * libname = "lib.so";
    struct link_map * map = nullptr;
    dlinfo(library, RTLD_DI_LINKMAP, &map);
    Elf64_Sym * symtab = nullptr;
    char * strtab = nullptr;
    int symentries = 0;
    for (auto section = map->l_ld; section->d_tag != DT_NULL; ++section)
    {
        if (section->d_tag == DT_SYMTAB)
        {
            symtab = (Elf64_Sym *)section->d_un.d_ptr;
        }
        if (section->d_tag == DT_STRTAB)
        {
            strtab = (char*)section->d_un.d_ptr;
        }
        if (section->d_tag == DT_SYMENT)
        {
            symentries = section->d_un.d_val;
        }
    }
    int size = strtab - (char *)symtab;
    for (int k = 0; k < size / symentries; ++k)
    {
        auto sym = &symtab[k];
        // If sym is function
        if (ELF64_ST_TYPE(symtab[k].st_info) == STT_FUNC)
        {
            //str is name of each symbol
            auto str = &strtab[sym->st_name];
            printf("%s\n", str);
        }
    }
OLD
I believe author does not need this anymore but maybe someone need actual code and here it is (based on previous answer)
First, we need a callback for dl_iterate_phdr():
static int callback(struct dl_phdr_info *info, size_t size, void *data)
{
    // data is copy of 2nd arg in dl_iterate_phdr
    // you can use it for your lib name as I did
    const char * libname = (const char *)data;
    // if current elf's name contains your lib
    if (strstr(info->dlpi_name, libname))
    {
        printf("loaded %s from: %s\n", libname, info->dlpi_name);
        for (int j = 0; j < info->dlpi_phnum; j++)
        {
            // we need to save dyanmic section since it contains symbolic table
            if (info->dlpi_phdr[j].p_type == PT_DYNAMIC)
            {
                Elf64_Sym * symtab = nullptr;
                char * strtab = nullptr;
                int symentries = 0;
                auto dyn = (Elf64_Dyn *)(info->dlpi_addr + info->dlpi_phdr[j].p_vaddr);
                for (int k = 0; k < info->dlpi_phdr[j].p_memsz / sizeof(Elf64_Dyn); ++k)
                {
                    if (dyn[k].d_tag == DT_SYMTAB)
                    {
                        symtab = (Elf64_Sym *)dyn[k].d_un.d_ptr;
                    }
                    if (dyn[k].d_tag == DT_STRTAB)
                    {
                        strtab = (char*)dyn[k].d_un.d_ptr;
                    }
                    if (dyn[k].d_tag == DT_SYMENT)
                    {
                        symentries = dyn[k].d_un.d_val;
                    }
                }
                int size = strtab - (char *)symtab;
                // for each string in table
                for (int k = 0; k < size / symentries; ++k)
                {
                    auto sym = &symtab[k];
                    auto str = &strtab[sym->st_name];
                    printf("%s\n", str);
                }
                break;
            }
        }
    }
    return 0;
}
Next, we call dl_iterate_phdr():
int main()
{
    auto library = dlopen("/path/to/library.so", RTLD_LAZY | RTLD_GLOBAL);
    const char * libname = "library.so";
    dl_iterate_phdr(callback, (void*)libname);
    return 0;
}
If you need to store those names somewhere, you can pass a pointer to your container, restore it with cast and write there.
For my example library:
#include "simple_lib.h"
#include <cstdio>
void __attribute__ ((constructor)) initLibrary(void)
{
    printf("Library is initialized\n");
}
void __attribute__ ((destructor)) cleanUpLibrary(void)
{
    printf("Library is exited\n");
}
void make_number()
{
    printf("1\n");
}
Prints this:
Library is initialized
_ITM_deregisterTMCloneTable
puts
__gmon_start__
_ITM_registerTMCloneTable
__cxa_finalize
_Z11initLibraryv
make_number
_Z14cleanUpLibraryv
Library is exited